Sun-Sig Byun's research while affiliated with Seoul National University and other places

Publications (153)

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This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and W2,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}...
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Optimal regularity estimates are established for the gradient of solutions to non-uniformly elliptic equations of Orlicz double phase with variable exponents type in divergence form under sharp conditions on such highly nonlinear operators for the Calderón–Zygmund theory.
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In this paper, we study the existence of distributional solutions solving (1.3) on a bounded domain Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{docum...
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We establish a global Calderón–Zygmund theory for the weak solution of the following p-Laplacian system $$\begin{aligned} {\left\{ \begin{array}{ll} u_t-{{\,\mathrm{div}\,}}\left(a(x,t)|\nabla u|^{p-2}\nabla u\right)=-{{\,\mathrm{div}\,}}|F|^{p-2}F+f&{}\text { in }\Omega _T,\\ u=0&{}\text { on }\partial \Omega \times (0,T),\\ u=u_0&{}\text { on }\O...
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We consider a nonlinear elliptic equation with Orlicz growth having a diffusive measure on the right-hand side. A maximal gradient integrability for such a measure data problem is established in the scale of Marcinkiewicz–Morrey spaces.
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This paper is concerned with a borderline case of double phase problems with a finite Radon measure on the right-hand side. We obtain sharp fractional regularity estimates for such non-uniformly elliptic problems.
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We investigate a general class of the so-called double phase problems on bounded domains. We establish an optimal global Calderón-Zygmund theory for such a non-uniformly elliptic problem under the assumptions that the coefficients have a small BMO semi-norm and the domain is sufficiently flat. Our regularity results not only cover discontinuous coe...
Preprint
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We establish a gradient estimate for a very weak solution to a quasilinear elliptic equation with a nonstandard growth condition, which is a natural generalization of the $p$-Laplace equation. We investigate the maximum extent for the gradient estimate to hold without imposing any regularity assumption on the nonlinearity other than basic structure...
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In this paper we are concerned with global maximal regularity estimates for elliptic equations with degenerate weights. We consider both the linear case and the non-linear case. We show that higher integrability of the gradients can be obtained by imposing a local small oscillation condition on the weight and a local small Lipschitz condition on th...
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We study generalized fractional $p$-Laplacian equations to prove local boundedness and H\"older continuity of weak solutions to such nonlocal problems by finding a suitable fractional Sobolev-Poincar\'e inquality.
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An irregular obstacle problem with a non-uniformly elliptic operator in divergence form of (G,H)-growth is studied. We provide local Calderón-Zygmund type estimates for an Orlicz double phase problem by proving that the gradient of a solution is integrable as both the gradient of the obstacle function and the associated nonhomogeneous term in the d...
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We deal with general quasilinear divergence-form coercive operators whose prototype is the [Formula: see text]-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is suppo...
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We prove local boundedness and H\"older continuity for weak solutions to nonlocal double phase problems concerning the following fractional energy functional \[ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}} + a(x,y)\frac{|v(x)-v(y)|^q}{|x-y|^{n+tq}}\, dxdy, \] where $0<s\le t<1<p \leq q<\infty$ and $a(\cdot,\cdot) \geq 0...
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In this paper, we study quasilinear parabolic equations with the nonlinearity structure modeled after the p(x, t)-Laplacian on nonsmooth domains. The main goal is to obtain end point Calderón-Zygmund type estimates in the variable exponent setting. In a recent work [1], the estimates obtained were strictly above the natural exponent p(x, t) and hen...
Article
We study quasilinear elliptic double obstacle problems with a variable exponent growth when the right-hand side is a measure. A global Calderón-Zygmund estimate for the gradient of an approximable solution is obtained in terms of the associated double obstacles and a given measure, identifying minimal requirements for the regularity estimate.
Article
We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calderón-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique boundary problem is as integrable as the nonhomogeneous term in Lp spaces under minimal regularity requirement on the no...
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We provide comprehensive regularity results and optimal conditions for a general class of functionals involving Orlicz multi-phase of the type \begin{align} \label{abst:1} v\mapsto \int_{\Omega} F(x,v,Dv)\,dx, \end{align} exhibiting non-standard growth conditions and non-uniformly elliptic properties. The model functional under consideration is giv...
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We prove optimal gradient estimates for distributional solutions to non-uniformly elliptic equations of multi-phase type in divergence form by investigating sharp conditions on such nonlinear operators for the Calderón-Zygmund theory.
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We investigate the existence of distributional solutions to nonlinear elliptic problems with general growth when the right-hand side is a bounded Radon measure. An optimal global Calderón-Zygmund type estimate for such problems is obtained by using mapping property of the fractional maximal function of the measure.
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We provide an optimal global Calderón-Zygmund theory for quasilinear elliptic equations of a very general form with Orlicz growth on bounded nonsmooth domains under minimal regularity assumptions of the nonlinearity A = A ( x , u , D u ) A=A(x,u,Du) in the first and second variables ( x , z ) (x,z) as well as on the boundary of the domain. Our resu...
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We prove maximal differentiability for the gradient of solutions to a certain type of nonlinear elliptic equations with a measure on the right-hand side. Our results generalize the limiting case of Calderón–Zygmund theory for an elliptic equation with $p$-growth to the case of Orlicz growth.
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We consider degenerate and singular parabolic equations with p-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (in...
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We study a nonlinear parabolic equation with a finite Radon measure on the right-hand side. An optimal regularity assumption on the coefficients is identified to obtain a sharp fractional differentiability for such a parabolic measure data problem.
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We are concerned with an optimal regularity for ω-minimizers to double phase variational problems with variable exponents where the associated energy density is allowed to be discontinuous. We identify basic structure assumptions on the density for the absence of Lavrentiev phenomenon and higher integrability. Moreover, we establish a local Calderó...
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We consider nonlinear elliptic measure data problems having coefficients in Cγα with α∈(0,1] and γ∈[1,∞). An optimal regularity assumption on the coefficients is investigated to obtain fractional differentiability results in a completely linearized form.
Preprint
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We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calder\'{o}n-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique boundary problem is as integrable as the nonhomogeneous term in $L^{p}$ spaces under minimal regularity requirement...
Preprint
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This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and $W^{2,p}$-regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.
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We establish a sharp higher integrability near the initial boundary for a weak solution to the following p -Laplacian type system: \left\{\begin{aligned} \displaystyle u_{t}-\operatorname{div}\mathcal{A}(x,t,% \nabla u)&\displaystyle=\operatorname{div}\lvert F\rvert^{p-2}F+f&&% \displaystyle\phantom{}\text{in}\ \Omega_{T},\\ \displaystyle u&\displa...
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We prove Calderón-Zygmund type estimates for distributional solutions to non-uniformly elliptic equations of generalized double phase type in divergence form. In particular, we provide sharp conditions on the nonlinear operators to establish the Calderón-Zygmund type estimates.
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We develop a global Calderón–Zygmund theory for a quasilinear divergence form parabolic operator with discontinuous entries which exhibit nonlinearities both with respect to the weak solution u and its spatial gradient Du in a nonsmooth domain. The nonlinearity behaves as the parabolic p-Laplacian in Du, its discontinuity with respect to the indepe...
Article
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In this paper we study a double phase problem with an irregular obstacle. The energy functional under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm, which can be regarded as a borderline case of the double phase functional with (p,q)‐growth. We obtain an optim...
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We study a nonlinear elliptic double phase problem with variable exponents to prove Calderón-Zygmund estimates under minimal regularity requirements on the nonlinearities.
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We obtain a global Calderón-Zygmund estimate for a borderline case of double phase problems with measure data in terms of the 1-fractional maximal function of the measure.
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We study an irregular double obstacle problem with Orlicz growth over a nonsmooth bounded domain. We establish a global Calderón–Zygmund estimate by proving that the gradient of the solution to such a nonlinear elliptic problem is as integrable as both the nonhomogeneous term in divergence form and the gradient of the associated double obstacles. W...
Preprint
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We study quasilinear elliptic double obstacle problems with a variable exponent growth when the right-hand side is a measure. A global Calder\'{o}n-Zygmund estimate for the gradient of an approximable solution is obtained in terms of the associated double obstacles and a given measure, identifying minimal requirements for the regularity estimate.
Article
We study regularity results for nonlinear parabolic systems of p-Laplacian type with inhomogeneous boundary and initial data, with \(p\in (\frac{2n}{n+2},\infty )\). We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In...
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We study fully nonlinear elliptic equations with oblique boundary conditions. We obtain a global W2,p-estimate, n−τ0<p<∞, for viscosity solutions of such problems when the boundary of the domain is in C2,α for every 0<α<1.
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We prove a global Calderón–Zygmund type estimate for the gradient of a solution to a nonlinear elliptic problem with nonstandard growth when the right-hand side is a bounded Radon measure. Minimal regularity requirements on both the nonlinearity and the boundary of the domain are investigated for such a gradient estimate.
Preprint
We obtain Calder\'on-Zygmund type estimates in generalized Morrey spaces for nonlinear equations of $p$-Laplacian type. Our result is obtained under minimal regularity assumptions both on the operator and on the domain. This result allows us to study asymptotically regular operators. As a byproduct, we obtain also generalized H\"older regularity of...
Preprint
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We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (...
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We study a generalized Stokes system with Orlicz growth which is nonstandard in a non-smooth domain. Our purpose is to derive a Calderon-Zygmund type estimate of the gradient of a solution and the pressure to such a system like (1.1) under a small BMO non-linearity and a sufficient flatness on the boundary of the domain. In the process, we overcome...
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We prove gradient Riesz potential estimates for nonlinear elliptic systems with subquadratic growth of the form −div(A(x,Du))=f, adopting ε-regularity criteria to the non-homogeneous setting. Assuming Dini type continuity in the first variable of the vector field A, we obtain partial C ¹ -regularity of solutions to such problems via Riesz potential...
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In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the duality pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the compactness argument developed in \cite{bulicek2018well}. In order to obtain the a priori estimates, we make use of th...
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An elliptic double phase problem with irregular double obstacles is investigated to establish a Calderón-Zygmund type estimate in the setting of Lebesgue spaces and weighted Lebesgue spaces. We prove that the gradient of a solution to such a highly nonlinear problem is as integrable as both the nonhomogeneous term in divergence form and the gradien...
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We provide Calderón–Zygmund estimates for ω-minimizers of double phase variational problems.
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This paper investigates the higher integrability in homogenization theory for a generalized steady state Stokes system in divergence form with discontinuous coefficients in a bounded nonsmooth domain. We obtain a global and uniform Calderón–Zygmund estimate by essentially proving that both the gradient of the weak solution and its associated pressu...
Article
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We study a nonlinear elliptic equation with measurable nonlinearity in a nonsmooth domain when the righthand side is a measure. A global Calderón-Zygmund type estimate in variable exponent spaces is established under an optimal regularity assumption on the nonlinearity and the Reifenberg flatness of the boundary.
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We investigate spherical quasi-minimizers (or so-called spherical Q-minimizers) u for integral functionals with p(x)-growth of F(u,Ω)≔∫Ωf(x,Du)−|G|p(x)−2G⋅Dudx.In particular, we find minimal regularity requirements on the integrand f(x,ξ), the variable exponent p(x), the boundary ∂Ω of the bounded domain Ω as well as Q under which |G|p(⋅)∈Lq(Ω)⟹|Du...
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We prove the natural weighted Calder\'{o}n and Zygmund estimates for solutions to elliptic and parabolic obstacle problems in nondivergence form with discontinuous coefficients and irregular obstacles. We also obtain Morrey regularity results for the Hessian of the solutions and H\"{o}lder continuity of the gradient of the solutions.
Preprint
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In this paper, we study the existence of distributional solutions solving \cref{main-3} on a bounded domain $\Omega$ satisfying a uniform capacity density condition where the nonlinear structure $\mathcal{A}(x,t,\nabla u)$ is modelled after the standard parabolic $p$-Laplace operator. In this regard, we need to prove a priori estimates for the grad...
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We study regularity results for nonlinear parabolic systems of $p$-Laplacian type with inhomogeneous boundary and initial data, with $p\in(\frac{2n}{n+2},\infty)$. We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In p...
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We study a nonlinear elliptic double obstacle problem with irregular data and establish an optimal Calderón–Zygmund theory. The partial differential operator is of the p-Laplacian type and includes merely measurable coefficients in one variable. We prove that the gradient of a weak solution is as integrable as both the gradient of assigned two obst...
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We derive global gradient estimates for $W^{1,p}_0(\Omega)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and H\"older...
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We consider a nonlinear parabolic equation with measurable nonlinearity in a nonsmooth bounded domain when the right-hand side is a finite signed Radon measure. Under optimal regularity assumptions on the nonlinearity and the boundary of the domain, we prove a global Calderón–Zygmund type estimate in weighted Orlicz spaces. As an application we obt...
Preprint
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In this paper, we study quasilinear parabolic equations with the nonlinearity structure modeled after the $p(x,t)$-Laplacian on nonsmooth domains. The main goal is to obtain end point Calder\'on-Zygmund type estimates in the variable exponent setting. In a recent work \cite{byun2016nonlinear}, the estimates obtained were strictly above the natural...
Article
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In this paper, we obtain weighted norm inequalities for the spatial gradients of weak solutions to quasilinear parabolic equations with weights in the Muckenhoupt class $A_{\frac{q}{p}}(\mathbb{R}^{n+1})$ for $q\geq p$ on non-smooth domains. Here the quasilinear nonlinearity is modelled after the standard $p$-Laplacian operator. Until now, all the...
Article
An irregular obstacle problem with non-uniformly elliptic operator in divergence form of (p,q)-growth is studied. We find an optimal regularity for such a double phase obstacle problem by essentially proving that the gradient of a solution is as integrable as both the gradient of the assigned obstacle function and the associated nonhomogeneous term...
Article
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We prove boundary higher integrability for the (spatial) gradient of \emph{very weak} solutions of quasilinear parabolic equations of the form $$ \left\{ \begin{array}{ll} u_t - div \mathcal{A}(x,t,\nabla u) = 0 &\quad \text{on} \ \Omega \times (-T,T), \\ u = 0 &\quad \text{on} \ \partial \Omega \times (-T,T), \end{array} \right. $$ where the non-l...
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We prove boundary higher integrability for the (spatial) gradient of \emph{very weak} solutions of quasilinear parabolic equations of the form $$u_t - \text{div}\,\mathcal{A}(x,t, \nabla u)=0 \quad \text{on} \ \Omega \times \mathbb{R},$$ where the non-linear structure $\text{div}\,\mathcal{A}(x, t,\nabla u)$ is modelled after the $p$-Laplace operat...
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We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and H\"{o}lder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is H\"{o}lder continuous.
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We study an asymptotically regular problem of (Formula presented.)-Laplacian type with discontinuous nonlinearity in a nonsmooth bounded domain. A global Calderón–Zygmund estimate is established for such a nonlinear elliptic problem with nonstandard growth under the assumption that the associated nonlinearity has a more general kind of the asymptot...
Article
In this paper we develop a global W2,p estimate for the viscosity solution of the Dirichlet problem of fully nonlinear elliptic equations F(D2u,Du,u,x)=f(x) in Ω,u=0 on ∂Ω to a more general function space. Given an N-function Φ and a Muckenhoupt weight w, we prove that if f belongs to the associated weighted Orlicz space LwΦ(Ω), then D2u∈LwΦ(Ω) and...
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A double phase problem with a bounded Borel measure on the right hand side is studied. We prove an optimal pointwise gradient estimate for such a measure data problem via Riesz potentials of the measure under log-Dini continuity assumption on the modulating coefficients. As a consequence, we find an optimized C1 regularity criterion.
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We establish the global Morrey regularity and continuity results for solutions to nonlinear elliptic equations over bounded nonsmooth domains. The novelty of our contribution is that the principal part of the operator is assumed to be merely asymptotically regular with respect to the gradient of a solution, which means that it behaves like the p-La...
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We prove the global Calderón-Zygmund estimates for second order parabolic equations in nondivergence form in weighted variable exponent Lebesgue spaces. We assume that the associated variable exponent is log-Hölder continuous, the weight is of a certain Muckenhoupt class with respect to the variable exponent, the coefficients of the equation are th...
Article
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In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the $p(x)$-Laplacian on nonsmooth domains and obtain sharp Calder\'on-Zygmund type estimates in the variable exponent setting. In a recent work of \cite{BO}, the estimates obtained were strictly above the natural exponent and hence there was a gap between th...
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In this paper we study the following singular p(x)-Laplacian problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} - \text{ div } \left( |\nabla u|^{p(x)-2} \nabla u\right) =\frac{ \lambda }{u^{\beta (x)}}+u^{q(x)}, &{} \text{ in }\quad \Omega , \\ u>0, &{} \text{ in }\quad \Omega , \\ u=0, &{} \text{ on }\quad \partial \Omega , \end{array}\...
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We consider a nonlinear and non-uniformly elliptic problem in divergence form on a bounded domain. The problem under consideration is characterized by the fact that its ellipticity rate and growth radically change with the position, which provides a model for describing a feature of strongly anisotropic materials. We establish the global Calderón–Z...
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We establish a local Lipschitz regularity near the boundary of weak solutions to a general class of homogeneous quasilinear elliptic equations with Neumann boundary condition in bounded convex domains.
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We consider a double phase problem with BMO coefficient in divergence form on a bounded nonsmooth domain. The problem under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm according to the position, which describes a feature of strongly anisotropic materials. We...
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We introduce a parabolic analogue of Muckenhoupt weights to study optimal weighted regularity in Orlicz spaces for a general nonlinear parabolic problem of p-Laplacian-type in divergence form over a nonsmooth domain. Assuming that the nonlinearity is measurable with respect to the time variable and has a small bounded mean oscillation (BMO) with re...
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We deal with the Dirichlet problem for general quasilinear elliptic equations over Reifenberg flat domains. The principal part of the operator supports natural gradient growth and its x-discontinuity is of small-BMO type, while the lower order terms satisfy controlled growth conditions with x-behaviour modelled by Morrey spaces. We obtain a Calderó...
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We investigate a quasilinear elliptic equation with variable growth in a bounded nonsmooth domain involving a signed Radon measure. We obtain an optimal global Calderón-Zygmund type estimate for such a measure data problem, by proving that the gradient of a very weak solution to the problem is as globally integrable as the first order maximal funct...
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We establish a global Calderón-Zygmund estimate for homogenization of a parabolic system in divergence form with discontinous coefficients have small BMO seminorms and the boundary of the domain is δ-flat for some δ < 0 depending on the given data.
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We prove an optimal theorem for a weak solution of an elliptic system in divergence form with measurable coefficients in a homogenization problem. Our theorem is sharp with respect to the assumption on the coefficients. Indeed, we allow the very rapidly oscillating coefficients to be merely measurable in one variable.
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We prove boundedness of the weak solutions to the Cauchy–Dirichlet problem for quasilinear parabolic equations whose prototype is the parabolic m-Laplacian. The nonlinear terms satisfy sub-controlled growth conditions with respect to the unknown function and its spatial gradient, while the behaviour in the independent variables is modelled in Lebes...
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We consider a generalized steady Stokes system with discontinuous coefficients in a nonsmooth domain when the inhomogeneous term belongs to a weighted $L^q$ space for $2<q<\infty$. We prove the global weighted $L^q$-estimates for the gradient of the weak solution and an associated pressure under the assumptions that the coefficients have small BMO...
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We study homogenization of the conormal derivative problem for an elliptic system with discontinuous coefficients in a bounded domain. A uniform global (Formula presented.) estimate for (Formula presented.) is obtained under optimal assumptions that the coefficients have a small bounded mean oscillation (BMO) seminorm and the domain is a (Formula p...
Article
A generalized elliptic equation with nonstandard growth in a nonsmooth bounded domain is studied. A global nonlinear gradient estimate is obtained in the frame of Orlicz spaces under the assumptions that the associated Young functions satisfy some moderate growth and decay conditions and that the boundary of the domain is -Reifenberg flat with depe...
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We study the conormal derivative problem for an elliptic equation of p-Laplacian type with discontinuous coefficients in a non-smooth domain in order to look for the minimal assumptions necessary to have the nonlinear Calderón-Zygmund theory for such problem. Under the assumptions that the nonlinear operator is sufficiently close to the p-Laplacian...
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We study nonlinear elliptic equations of p(x)-Laplacian type on nonsmooth domains to obtain an optimal Calderón-Zygmund type estimate in the variable exponent spaces. We find a correct regularity assumption on p((dot operator)), a minimal regularity requirement on the associated nonlinearity and a suitable flatness condition on the boundary of the...
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In this paper, we examine an elliptic system of divergence form in a homogenization problem in order to obtain a global (Formula presented.) estimate for (Formula presented.). Throughout our study, the coefficients are assumed to have a small bounded mean oscillation seminorm, and the boundary of the domain is assumed to be flat in the Reifenberg s...
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We introduce asymptotically elliptic equations and establish the global estimates for viscosity solutions to both asymptotically linear elliptic equations and asymptotically fully nonlinear elliptic equations in bounded domains. Our approach for the proof is based on an appropriate transformation which converts a given asymptotically elliptic equat...
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We study elliptic equations with measurable nonlinearities in nonsmooth domains. We establish an optimal global W1,p estimate under the condition that the associated nonlinearity is allowed to be merely measurable in one variable but has a sufficiently small BMO semi-norm in the other variables, while the underlying domain is sufficiently flat in t...
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We study the global regularity in generalized Morrey spaces of the solutions to variational inequality and obstacle problem related to divergence form parabolic operator in bounded non-smooth domain. We impose minimal regularity conditions as to the coefficients of the operator so also to the boundary of the domain.
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We study the Calderón-Zygmund theory for nonlinear parabolic problems in the setting of the variable exponent Lebesgue spaces. In particular, we prove the global Ls(.) integrability of the gradient of solutions to parabolic equations with p(.) growth in nonsmooth domains with s(.) > p(.). In addition, we present precise regularity conditions on the...
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We prove global regularity in weighted Lebesgue spaces for the viscosity solutions to the Dirichlet problem for fully nonlinear elliptic equations. As a consequence, regularity in Morrey spaces of the Hessian is derived as well.
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A global Calderón-Zygmund estimate type estimate in Weighted Lorentz spaces and Lorentz-Morrey spaces is obtained for weak solutions to elliptic obstacle problems of p-Laplacian type with discontinuous coefficients over Reifenberg flat domains.
Article
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We study a nonlinear elliptic problem with an irregular obstacle in a bounded nonsmooth domain when the nonlinearity is merely asymptotically regular. We find an optimal regularity requirement on the associated nonlinearity and a minimal geometric condition on the boundary to ensure a global Calderón–Zygmund estimate for such an asymptotically regu...
Article
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We consider nonhomogeneous elliptic and parabolic problems involving a discontinuous nonlinearity and an asymptotic regularity over an irregular domain in divergence form of $p$-Laplacian type, to establish the global Calderón–Zygmund estimate by converting a given asymptotically regular problem to a suitable regular problem.
Article
We obtain the global weighted Orlicz regularity of the maximum order derivatives of the solution to the Dirichlet problem for nondivergence parabolic equations in a bounded domain. We find correct conditions on the associated weight and Young function. The coefficients are assumed to have small bounded mean oscillation (BMO) seminorms.
Article
We establish an optimal $L^{p(\cdot )}$-regularity theory for the gradient of weak solutions for parabolic equations in divergence form with bounded measurable coefficients in rough domains beyond the Lipschitz category. With a function $p(\cdot )=p(x,t)$ of spatial and time variables satisfying log-Hölder continuity, we prove that the spatial grad...

Citations

... For a general overview, we point the reader to [4]. One of the appeals to study problems in generalized Orlicz spaces is to unify theory of many distinctly studied frameworks, such as polynomial, variable exponent, double phase, Orlicz and many others, see for example [1,3,8,12,13,30,33]. As already mentioned, existence and uniqueness have been studied widely in generalized Orlicz spaces, but properties of solutions are vigorously studied, too [2,5,6,24,28]. ...
... The seminal result of De Giorgi [7] and Nash [17] ensures boundedness and Hölder continuity of the W 1,2 0 -weak solutions to linear elliptic equations with L p -coefficients and it was later extended by Ladyzhenskaya and Ural'tseva [11] to the case of quasilinear equations. Recently, boundedness and Hölder continuity of the weak solutions to general quasilinear equations have been proved ( [3,4] when p = 2 and [5,6] when p ∈ (1, n]) allowing control in terms of Morrey spaces for the x-behaviour of the nonlinear terms. ...