Mahamadi Warma

George Mason University | GMU · Department of Mathematical Sciences

The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore–Gibson–Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. Assuming that b>0 and α-τc2b>0, we show that if 00. However, we prove that for 0

We introduce some families of generalized Black--Scholes equations which involve the Riemann-Liouville and Weyl space-fractional derivatives. We prove that these generalized Black--Scholes equations are well-posed in $(L^1-L^\infty)$-interpolation spaces. More precisely, we show that the elliptic type operators involved in these equations generate...

This chapter provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps, and deep (machine) learning. Various notions of solutions to linear fractional elliptic equation...

The aim of this chapter is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other rele...

In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control prob...

We consider parabolic equations on bounded smooth open sets \begin{document}$ {\Omega}\subset \mathbb{R}^N $\end{document} (\begin{document}$ N\ge 1 $\end{document}) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator \begin{document}$ \mathscr{L} : = - \Delta + (-\Delta)^{s} $\end{document} (\begin{document...

We consider optimal control of fractional in time (subdiffusive, i.e., for 0 < γ < 1) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we first show the existence and regularity of solutions to the forward and the associated backward (adjoint) prob...

We consider optimal control of fractional in time (subdiffusive, i.e., for $% 0<\gamma <1$) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we{~\textsf{first show}} the existence and regularity of solutions to the forward and the associated {\text...

This article provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations...

The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls. Our reference model will be a non-local diffu...

In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control prob...

We consider parabolic equations on bounded smooth open sets $\Om\subset \R^N$ ($N\ge 1$) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator $\mathscr{L} \coloneqq - \Delta + (-\Delta)^{s}$ ($0<s<1$). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic pro...

Let $A$ be a densely defined closed, linear $\omega$-sectorial operator of angle $\theta\in [0,\frac{\pi}{2})$ on a Banach space $X$ for some $\omega\in\mathbb R$. We give an explicit representation (in terms of some special functions) and study the precise asymptotic behavior as time goes to infinity of solutions to the following diffusion equatio...

We investigate the controllability from the exterior of space-time fractional wave equations involving the Caputo time fractional derivative with the fractional Laplace operator subject to nonhomogeneous Dirichlet or Robin type exterior conditions. We prove that if 1 < α< 2 , 0 < s< 1 and Ω ⊂ RN is a bounded Lipschitz domain, then the system {Dtαu+...

Let \(\varOmega \subset {{\mathbb {R}}}^n\) (\(n\ge 1\)) be a bounded open set with a Lipschitz continuous boundary. In the first part of the paper, using the method of bilinear forms we give a characterization of the realization in \(L^2(\varOmega )\) of the fractional Laplace operator \((-\Delta )^s\) (\(0<s<1\)) with the nonlocal Neumann and Rob...

We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator AB with compact resolvent on L2(Ω), where \({\Omega }\subset \mathbb {R}^{N}\) (N ≥ 1) is a bounded open set. More precisel...

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question...

In this chapter, we consider some general classes of reaction–diffusion systems that contain some fractional kinetics occurring in applications, and then investigate their local and global existence of solutions in detail. In a preliminary step, we derive results that allow for the existence of sufficiently smooth solutions which are needed in orde...

In the present chapter, we rely on the crucial results of Chap. 2 to develop well-posedness results in the same spirit of Rothe where second order elliptic operators in divergence form have been considered for the classical parabolic problem (α = 1).

In this monograph, we first consider a semilinear fractional kinetic equation that is characterized by the presence of a nonlinear time-dependent source \(f=f\left ( x,t,u\right ) \), a generalized time derivative \(\partial _{t}^{\alpha }\) in the sense of Caputo and the presence of a large class of diffusion operators A.

We first introduce some background. Let Y, Z be two Banach spaces endowed with norms \(\left \Vert \cdot \right \Vert { }_{Y}\) and \(\left \Vert \cdot \right \Vert { }_{Z}\), respectively. We denote by Y ↪Z if Y ⊆ Z and there exists a constant C > 0 such that \(\left \Vert u\right \Vert { }_{Z}\leq C\left \Vert u\right \Vert { }_{Y},\) for u ∈ Y ⊆...

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Control problems play a key role in engineering, economics and sciences. To be more precise, in climate sciences, often times, relevant p...

We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation associated with the fractional Laplace operator subject to the non-homogeneous Dirichlet type exterior condition. In the first part, we show that if 0 < s < 1, Ω ⊂ ℝ N ( N ≥ 1) is a bounded Lipschitz domain and the paramete...

This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic partial differential equations. Several mathematical tools are developed during the process to study these problems, for instance, the characterization of the dual of fractional-order Sobolev spaces and the well-posedness of fractional...

This paper considers optimal control of fractional parabolic PDEs with both state and control constraints. The key challenge is how to handle the state constraints. Similarly, to the elliptic case, in this paper, we establish several new mathematical tools in the parabolic setting that are of wider interest. For example, existence of solution to th...

We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator $A_B$ with a compact resolvent on $L^2(\Omega)$, where $\Omega\subset\mathbb{R}^N$ ($N\ge 1$) is a bounded open set. More p...

In \cite{HAntil_RKhatri_MWarma_2018a} we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been develo...

In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian (−dx ²)s (0 < s < 1) on the interval (−1, 1). We prove the existence of a minimal (strictly positive) time Tmin such that the fractional heat dynamics can be controll...

In a bounded domain, we consider a thermoelastic plate with rotational forces. The rotational forces involve the spectral fractional Laplacian, with power parameter $0\le\ta\le 1$. The model includes both the Euler-Bernoulli ($\ta=0$) and Kirchhoff ($\ta=1$) models for thermoelastic plate as special cases. First, we show that the underlying semigro...

We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator $ (-\partial_x^2)^s$ (with $0<s<1$) on the interval $(-1,1)$. Our control function is localized in an open set $\mathcal O$ in the exterior of $(-1,1)$, that is, $\mathc...

Let $(-\Delta)_c^s$ be the realization of the fractional Laplace operator on the space of continuous functions $C_0(\mathbb{R})$, and let $(-\Delta_h)^s$ denote the discrete fractional Laplacian on $C_0(\mathbb{Z}_h)$, where $0<s<1$ and $\mathbb{Z}_h:=\{hj:\; j\in\mathbb{Z}\}$ is a mesh of fixed size $h>0$. We show that solutions of fractional orde...

We prove representation results for solutions of a time‐fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem , where 0<β ≤ 2,...

Let \(\Omega \subset {\mathbb {R}}^N\) be an arbitrary open set, \(0<s<1\) and denote by \((e^{-t(-\Delta )_{{{\mathbb {R}}}^N}^s})_{t\ge 0}\) the semigroup on \(L^2({{\mathbb {R}}}^N)\) generated by the fractional Laplace operator. In the first part of the paper, we show that if T is a self-adjoint semigroup on \(L^2(\Omega )\) satisfying a fracti...

Very recently Warma (2019 SIAM J. Control Optim. to appear) has shown that for nonlocal PDEs associated with the fractional Laplacian, the classical notion of controllability from the boundary does not make sense and therefore it must be replaced by a control that is localized outside the open set where the PDE is solved. Having learned from the ab...

Abstract We prove the existence and uniqueness of a weighted pseudo asymptotically mild solution to the following class of abstract semilinear difference equations: u(n+1)=A∑k=−∞na(n−k)u(k+1)+∑k=−∞nb(n−k)f(k,u(k)),n∈Z, $$ u(n+1)= A \sum_{k=-\infty }^{n} a(n-k)u(k+1)+ \sum _{k=-\infty }^{n} b(n-k)f\bigl(k,u(k)\bigr),\quad n\in \mathbb{Z}, $$ where A...

This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic PDEs of order $s \in (0,1)$. There are several mathematical tools that are developed during the process to study this problem, for instance, the characterization of the dual of the fractional order Sobolev spaces and well-posedness of...

In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian $(-\Delta)^s$ ($0<s<1$) on the interval $(-1,1)$. We prove the existence of a minimal (strictly positive) time $T_{\rm min}$ such that the fractional heat dynamics ca...

Let $\Omega\subset\RR^n$ ($n\ge 1$) be a bounded open set with a Lipschitz continuous boundary. In the first part of the paper, using the method of bilinear forms, we give a rigorous characterization of the realization in $L^2(\Omega)$ of the fractional Laplace operator $(-\Delta)^s$ ($0<s<1$) with the nonlocal Neumann and Robin exterior conditions...

We derive optimal well-posedness results and explicit representations of solutions in terms of special functions for the linearized version of the equation (*){Dtβu(n,t)=−(−Δd)αu(n,t)+f(n−ct,u(n,t)),n∈Z,t>0, 0<α,β<1, u(n,0)=φ(n), n∈Z, for some constant c≥0, where Dtβ denotes the Caputo fractional derivative in time of order β and (−Δd)α denotes the...

In this paper we introduce a new notion of optimal control, or source identification in inverse, problems with fractional parabolic PDEs as constraints. This new notion allows a source/control placement outside the domain where the PDE is fulfilled. We tackle the Dirichlet, the Neumann and the Robin cases. For the fractional elliptic PDEs this has...

Let $\Omega\subset\R^N$ be an arbitrary open set and denote by $(e^{-t(-\Delta)_{\RR^N}^s})_{t\ge 0}$ (where $0<s<1$) the semigroup on $L^2(\RR^N)$ generated by the fractional Laplace operator. In the first part of the paper we show that if $T$ is a self-adjoint semigroup on $L^2(\Omega)$ satisfying a fractional Gaussian estimate in the sense that...

The implication $(i)\Rightarrow (ii)$ of Theorem 2.1 in our article [1] is not true as it stands. We give here two correct statements which follow from the original proof.

We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibr...

Let Ω ⊂ RN be a bounded domain with a Lipschitz continuous boundary. We study the controllability of the space-time fractional diffusive equation {Dα tu+(-Δ)s u = 0 in (0, T)×Ω,u = gχ(0,T )× O in (0, T) × (RN \Ω ),u(0,. ) = u0 in Ω}, where u = u(t, x) is the state to be controlled and g = g(t, x) is the control function which is localized in a none...

We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibr...

We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval (−1, 1), associated with the fractional Laplace operator (−∂²x)s, where 0 < s < 1. We show that there is a control function, which is localized in a nonempty open set O ⊂ (R \ (−1, 1)), that is, at the exterior of the interval (−1, 1...

The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore--Gibson--Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition.

We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval $(0,1)$ associated with the fractional Laplace operator $(-\partial_x^2)^s$, where $0<s<1$. We show that there is a control function which is localized in a non-empty open set $\mathcal{O}\subset \left(\mathbb{R}\setminus(0,1)\right)...

Very recently M. Warma has shown that for nonlocal PDEs associated with the fractional Laplacian, the classical notion of controllability from the boundary does not make sense and therefore must be replaced by a control that is localized outside the open set where the PDE is solved. Having learned from the above mentioned result in this paper we in...

We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. In the first part, we show that if $0<s<1$, $\Omega\subset\RR^N$ ($N\ge 1$) is a bounded Lipschitz domain and the paramet...

In this paper we investigate the following fractional order in time Cauchy problem [Formula presented]The fractional in time derivative is taken in the classical Caputo sense. In the scientific literature such equations are sometimes dubbed fractional-in time wave equations or super-diffusive equations. We obtain results on existence and regularity...

Let \(p\in (1,\infty )\) and \(\Omega \subset \mathbb {R}^{N}\) a bounded open set with Lipschitz continuous boundary \(\partial \Omega \). We define a fractional p-Dirichlet-to-Neumann operator associated with the regional fractional p-Laplace operator \((-\Delta )_{p,\Omega }^{s}\), \(0<s<1\), and prove that it generates a strongly continuous sem...

Let $\Om\subset\RR^N$ a bounded domain with a Lipschitz continuous boundary. We study the controllability of the space-time fractional diffusion equation \begin{equation*} \begin{cases} \mathbb D_t^\alpha u+(-\Delta)^su=0\;\;&\mbox{ in }\;(0,T)\times\Omega\\ u=g &\mbox{ in }\;(0,T)\times(\RR^N\setminus\Omega)\\ u(0,\cdot)=u_0&\mbox{ in }\;\Omega, \...

In this paper we study optimal control problems with the regional fractional p-Laplace equation, of order s ∈ (0, 1) and p ∈ [2, ∞), as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the regional fractional p-Laplace operator. We show...

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set \(\Omega \subset {\mathbb {R}}^N\). Proofs combine classical abstract regularity results for parabolic equations with some new local regularity resu...

In this paper we consider the optimal control of semilinear fractional PDEs with both spectral and integral fractional diffusion operators of order $2s$ with $s \in (0,1)$. We first prove the boundedness of solutions to both semilinear fractional PDEs under minimal regularity assumptions on domain and data. We next introduce an optimal growth condi...

This book provides a unified analysis and scheme for the existence and uniqueness of strong and mild solutions to certain fractional kinetic equations. This class of equations is characterized by the presence of a nonlinear time-dependent source, generally of arbitrary growth in the unknown function, a time derivative in the sense of Caputo and the...

Given a bounded domain Ω⊂RN with a Lipschitz boundary ∂Ω and p,q∈(1,+∞) , we consider the quasilinear elliptic equation -Δpu+α1u=f in Ω complemented with the generalized Wentzell-Robin type boundary conditions of the form bx∇up-2∂nu-ρbxΔq,Γu+α2u=g on ∂Ω . In the first part of the article, we give necessary and sufficient conditions in terms of the...

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with Dirichlet boundary condition on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. The key tool consists in combining classical abstract regularity results for parabolic equations with some new local regularity resu...

We prove the Wloc2⁢s,p local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of ℝN. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space....

We consider a transmission problem consisting of two semilinear parabolic equations involving fractional diffusion operators of different orders in a general non-smooth setting with emphasis on Lipschitz interfaces and (local) transmission conditions along the interface. We give a unified framework for the existence and uniqueness of strong and mil...

In [1], for 1 < p < ∞ {1<p<\infty} , we proved the W loc 2 ⁢ s , p {W^{2s,p}_{\mathrm{loc}}} local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian ( - Δ ) s {(-\Delta)^{s}} on an arbitrary bounded open set of ℝ N {\mathbb{R}^{N}} . Here we make a more precise and rigorous statement. In fact, f...

In this paper we study optimal control problems with either fractional or regional fractional $p$-Laplace equation, of order $s$ and $p\in [2,\infty)$, as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the fractional or the regional fr...

Let $\Omega\subset\mathbb R^N$ be a bounded open set with Lipschitz continuous boundary $\Gamma$. Let $\gamma>0$, $\delta\ge 0$ be real numbers and $\beta$ a nonnegative measurable function in $L^\infty(\Gamma)$. Using some suitable Carleman estimates, we show that the linear heat equation $\partial_tu - \gamma\Delta u = 0$ in $\Omega\times(0,T)$ w...

We consider a degenerate parabolic equation associated with the fractional $% p $-Laplace operator $\left( -\Delta \right) _{p}^{s}$\ ($p\geq 2$, $s\in \left( 0,1\right) $) and a monotone perturbation growing like $\left\vert s\right\vert ^{q-2}s,$ $q>p$ and with bad sign at infinity as $\left\vert s\right\vert \rightarrow \infty $. We show the exi...

Using regularized resolvent families, we investigate the solvability of the fractional order inhomogeneous Cauchy problem D;αt u(t) = Au(t) + f(t); t > 0; 0 < α < 1; where Dαt is the Caputo fractional derivative of order α, A a closed linear operator on some Banach space X, f : [0, ∞) → X is a given function. We dene an operator family associated w...

In the very influential paper \cite{CS07} Caffarelli and Silvestre studied regularity of $(-\Delta)^s$, $0<s<1$, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea \cite{ST10} and Gal\'e, Miana and Stinga \cite{GMS13} gave several more abstract versions of this extension procedure. The purpose of this p...

We study the boundary controllability of fractional wave (also known as super diffusion) equations associated to the Caputo fractional derivative with a symmetric non-negative uniformly elliptic operator subject to the non-homogeneous Dirichlet- or Robin-type boundary conditions. Our results show that if (Formula presented.) and (Formula presented....

We consider nonlinear nonlocal boundary value problems associated with fractional operators (including the fractional p-Laplace and the regional fractional p-Laplace operators) and subject to general (fractional-like) boundary conditions on bounded domains with Lipschitz boundary. Under suitable conditions on the nonlinearities of our system, we es...

In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order $s \in (0,1)$. We identify minimal conditions on the nonlinear term and the source which leads to existence of weak solutions and uniform $L^\infty$-bound on the solutions. Next we realize the fractional Laplacian as a D...

We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of gene...

We investigate a class of semilinear parabolic and elliptic problems with fractional dynamic boundary conditions. We introduce two new operators, the so-called fractional Wentzell Laplacian and the fractional Steklov operator, which become essential in our study of these nonlinear problems. Besides giving a complete characterization of well-posedne...

We investigate the long term behavior in terms of finite dimensional global attractors and (global) asymptotic stabilization to steady states, as time goes to infinity, of solutions to a non-local semilinear reaction-diffusion equation associated with the fractional Laplace operator on non-smooth domains subject to Dirichlet, fractional Neumann and...

We study the interior approximate controllability of fractional wave equations with the fractional Caputo derivative associated with a non negative self-adjoint operator satisfying the unique continuation property. Some well-posedness and fine regularity properties of solutions to fractional wave and fractional backward wave type equations are also...

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acti...

Let p ∈ (1,∞), s ∈(0,1) and Ω ⊂ ℝN an arbitrary bounded open set. In the first part we consider the inverse Pdbls,p:=[(-Δ)p,Ωs]-1 of the fractional p-Laplace operator (-Δ)p,Ωs with the Dirichlet boundary condition. We show that in the singular case p ∈(1,2), the operator Pdbls,p is locally Lipschitz continuous on L∞(Ω) and that global Lipschitz con...

We consider a transmission problem consisting of a semilinear parabolic equation in a general non-smooth setting with emphasis on rough interfaces which bear a fractal-like geometry and nonlinear dynamic (possibly, nonlocal) boundary conditions along the interface. We give a unified framework for existence of strong solutions, existence of finite d...

We prove an integration by part formula for the regional fractional p-Laplace operator and characterize the associated fractional p-Neumann and fractional p-Robin type boundary conditions

Let Ω⊂ ℝN be a bounded open set with Lipschitz continuous boundary ∂Ω. We define a fractional Dirichlet-to-Neumann operator and prove that it generates a strongly continuous analytic and compact semigroup on L2(∂Ω) which can also be ultracontractive. We also use the fractional Dirichletto-Neumann operator to compare the eigenvalues of a realization...

Let ${\Omega\subset\mathbb{R}^N}$ be an arbitrary open set with boundary ${\partial \Omega, 1 < p < \infty}$ and let ${f\in L^q(\Omega)}$ for some q > N > 1. In the first part of the article, we show that weak solutions of the quasi-linear elliptic equation ${-{\rm div}(|\nabla u|^{p-2} \nabla u)+a(x)|u|^{p-2}u=f}$ in Ω with the nonlocal Ro...

Let Omega subset of R-N (N >= 2) be a bounded domain with a boundary partial derivative Omega of class C-2 and let alpha, beta be maximal monotone graphs in R-2 satisfying alpha(0) boolean AND beta (0) there exists 0. Given f is an element of L-1(Omega) and g is an element of L-1(partial derivative Omega), we characterize the existence and uniquene...

Let be an arbitrary open set with boundary a,Omega. Let and sa(0,1). In the first part of the article we give some useful properties of the fractional order Sobolev spaces. We define a relative (s,p)-capacity on with the fractional order Sobolev spaces, give its properties and its connection with the classical Bessel (s,p)-capacity and the Hausdorf...

We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator.

Let A be a uniformly elliptic operator in divergence form with bounded coefficients. We show that on a bounded domain Ω⊂ℝ N with Lipschitz continuous boundary ∂Ω, a realization of Au-β 1 (x,u) in C(Ω ¯) with the nonlinear general Wentzell boundary conditions [Au-β 1 (x,u)]| ∂Ω -Δ Γ u+∂ ν a u+β 2 (x,u)=0 on ∂Ω generates a strongly continuous nonline...

We show that on a bounded domain Ω⊂ℝ N with Lipschitz continuous boundary ∂Ω, weak solutions of the elliptic equation λu-Au=f in Ω with the boundary conditions -γΔ Γ u+∂ ν a u+βu=g on ∂Ω are globally Hölder continuous on Ω ¯. Here A is a uniformly elliptic operator in divergence form with bounded measurable coefficients, Δ Γ is the Laplace-Beltrami...

We consider nonlinear elliptic partial differential equations for quasilinear operators of the formA(u)=−div(a(x,u,∇u))+A0(x,u,∇u),x∈Ω, subject to fully nonlinear boundary conditions involving boundary operators of the form, for each β⩾0β⩾0,Bβ(u)=−βdivΓ(b(x,u,∇Γu))+B0(x,u,∇u,∇Γu),x∈∂Ω. The main goal of this paper is to give, under suitable assumpti...

We prove a Riesz type representation theorem for lower semi-continuous, monotone, local functionals on Cc(X)+Cc(X)+, where XX is a locally compact, separable, metric space.

Fractional calculus is a subject of great interest in many areas of mathematics, physics and sciences, including stochastic processes, mechanics, chemistry, and biology. We will call an operator A on a Banach space X ω-sectorial (ω∈ℝ) of angle θ if there exists θ∈[0,π/2) such that S θ :={λ∈ℂ∖{0}|arg(λ)|<θ+π/2}⊂ρ(A) (the resolvent set of A) and sup{...

We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem Dtαu(t)=Au(t)+f(t), t>0, where 0<α<1. When A is the generator of a C0-semigroup (T(t))t≥0 on a Banach space X, we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary c...

Let p ∈ (1, ∞) and let Ω ⊆ ℝ N be a bounded domain with Lipschitz continuous boundary. We characterize on L 2(Ω) all order-preserving semigroups that are generated by convex, lower semicontinuous, local functionals and are sandwiched between the semigroups generated by the p-Laplace operator with Dirichlet and Neumann boundary conditions. We show t...

We show that on a bounded domain Ω⊂ℝ N with Lipschitz continuous boundary ∂Ω, if 2N/(N+2)<p≤N-1, then weak solutions of the quasi-linear elliptic equation -Δ p u+|u| p-2 u+α 1 (x,u)=finΩ with the general Wentzell boundary conditions -Δ p,Γ u+|∇u| p-2 ∂ v u+|u| p-2 u+α 2 (x,u)=gweaklyon∂Ω, are globally Hölder continuous on Ω ¯. Here, Δ p and Δ p,Γ d...

We show that a realization of the Laplace operator Au :=u′′ with general nonlocal Robin boundary conditions α j u′(j)+β j u(j)+γ 1–j u(1 − j)=0, (j=0, 1) generates a cosine family on L p (0, 1) for every p Î [1,¥){p\,{\in}\,[1,\infty)}. Here α j , β j and γ j are complex numbers satisfying α 0, α 1 ≠ 0. We also obtain an explicit represent...

Let p∈(1,N), Ω⊂RN a bounded W1,p-extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d∈(N−p,N). We show in the first part that for every p∈[2N/(N+2),N)∩(1,N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L...

We show that the semigroup generated by the realization of the Laplace operator with Wentzell boundary conditions in a bounded smooth domain is analytic on $${L^1(\Omega) \oplus L^1(\partial \Omega)}$$ .

We characterize the Lq-solvability of a class of quasi-linear elliptic equations involving the p-Laplace operator with generalized nonlinear Robin type boundary conditions on bad domains. Some uniqueness results are also given.

Let p is an element of [2, N), Omega subset of R(N) an open set and let mu be a Borel measure on partial derivative Omega. Under some assumptions on Omega, mu, f, g and beta, we show that the quasi-linear elliptic equation with nonlinear inhomogeneous Robin-type boundary conditions { -Delta(p)u + c(x)vertical bar u vertical bar(p-2)u = f in Omega d...

We consider a quasi-linear parabolic equation with nonlinear dynamic boundary conditions occurring as generalizations of semilinear reaction-diffusion equations with dynamic boundary conditions and various other phase-field models, such as the isothermal Allen-Cahn equation with dynamic boundary conditions. We thus formulate a class of initial and...

Let Ω⊂RNΩ⊂RN be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H1H1-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by ∂u∂νdσ+β(x,u)dμ=0 on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup S...

Let Omega subset of R(N) be a bounded domain with a Lipschitz boundary and let sigma be the restriction to partial derivative Omega of the (N - 1)-dimensional Hausdorff measure. Let B : partial derivative Omega x R --> [0, +infinity] be sigma-measurable in the first variable and assume that for sigma-almost every x is an element of partial derivati...