András Domokos

András Domokos
California State University, Sacramento | CSUS · Department of Mathematics and Statistics

PhD

About

44
Publications
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329
Citations
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August 2004 - present
California State University, Sacramento
Position
  • Professor (Full)

Publications

Publications (44)
Article
In this paper we prove uniform convergence of approximations to p-harmonic functions by using natural p-mean operators on bounded domains of the Heisenberg group H which satisfy an intrinsic exterior corkscrew condition. These domains include Euclidean C1,1 domains.
Preprint
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In this paper we point out an interesting geometric structure of nonnegative metric curvature emerging from the hyperspaces of decomposable, non-locally connected homogeneous continua, where "smooth" and "non-smooth" partitions live together, similarly to the macroscopic and the quantum realms of the Universe.
Article
We present self-contained proofs of the stability of the constants in the volume doubling property and the Poincaré and Sobolev inequalities for Riemannian approximations in Carnot groups. We use an explicit Riemannian approximation based on the Lie algebra structure that is suited for studying nonlinear subelliptic partial differential equations....
Preprint
Full-text available
In this paper we prove uniform convergence of approximations to $p$-harmonic functions by using natural $p$-mean operators on bounded domains of the Heisenberg group $\mathbb{H}$ which satisfy an intrinsic exterior corkscrew condition. These domains include Euclidean $C^{1,1}$ domains.
Article
Let the vector fields \(X_1, \ldots , X_{6}\) form an orthonormal basis of \(\mathcal {H}\), the orthogonal complement of a Cartan subalgebra (of dimension 2) in \({{\,\mathrm{SU}\,}}(3)\). We prove that weak solutions u to the degenerate subelliptic p-Laplacian $$\begin{aligned} \Delta _{\mathcal {H},{p}} u(x)=\sum _{i=1}^{6} X_i^{*}\left( |\nabla...
Preprint
Full-text available
In this paper we introduce a new type of exponential map in semi-simple compact Lie groups, which is related to the sub-Riemannian geometry generated by the orthogonal complement of a Cartan subalgebra in a similar way to how the group exponential map is related to the Riemannian geometry.
Preprint
Full-text available
Let the vector fields $X_1, ... , X_{6}$ form an orthonormal basis of ${\mathcal H}$, the orthogonal complement of a Cartan subalgebra (of dimension $2$) in SU(3). We prove that weak solutions $u$ to the degenerate subelliptic $p$-Laplacian $$ \Delta_{\mathcal{H},{p}} u(x)=\sum_{i=1}^{6} X_i^{*}\left(|\nabla_{\hspace{-0.1cm} {\mathcal H}} u|^{p-2}X...
Article
Length spectra for Riemannian metrics have been well studied, while sub-Riemannian length spectra remain largely unexplored. Here we give the length spectrum for a canonical sub-Riemannian structure attached to any compact Lie group by restricting its Killing form to the sum of the root spaces. Surprisingly, the shortest loops are the same in both...
Article
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We prove a fixed point theorem for mappings f defined on conical shells F in \(\mathbb R^n\), where the image of f need not be a subset of F, nor even a subset of the cone that contains F. In this sense, our results extend, in \(\mathbb R^n\), Krasnosel’skiĭ’s well-known fixed point result on cones in Banach spaces (Krasnosel’skiĭ, Soviet Math Dokl...
Article
We propose to find algebraic characterizations of the metric projections onto closed, convex cones in reflexive, locally uniformly convex Banach spaces with locally uniformly convex dual.
Article
Full-text available
In this paper we will use the algebraic information encoded in the root system of a semi-simple, connected, compact Lie group to describe properties of sub-Riemannian geodesics. First we give an algebraic proof that all sub-Riemannian geodesics are normal. We then find characterizations and lengths of the Riemannian and sub-Riemannian geodesic loop...
Article
Let X be a real Hilbert space. We give necessary and sufficient algebraic conditions for a mapping \({F\colon X \to X}\) with a closed image set to be the metric projection mapping onto a closed convex set. We provide examples that illustrate the necessity of each of the conditions. Our characterizations generalize several results related to projec...
Article
We will study the connections between the elliptic and subelliptic versions of the Peter-Weyl and Plancherel theorems, in the case when the sub-Riemannian structure is generated naturally by the choice of a Cartan subalgebra. Along the way we will introduce and study the subelliptic Casimir operator associated to the subelliptic Laplacian.
Article
Full-text available
In this paper, we consider a natural subelliptic structure in semisimple, compact and connected Lie groups, and estimate the constant in the so-called subelliptic Friedrichs–Knapp–Stein inequality, which has implications in the regularity theory of p-energy minimizers.
Article
We study the interior regularity for nondegenerate p-harmonic functions in Grušin planes of higher order. The local boundedness of the horizontal gradient is obtained by running a subelliptic variant of the Moser iteration scheme.
Article
In this paper, we study the higher order regularity for weak solutions of a class of quasilinear subelliptic equations. We introduce the notion of ν-closed Hörmander system of vector fields, which includes all the previously studied nilpotent systems and extends them to some classes of non-nilpotent systems of vector fields, including those generat...
Chapter
We prove C⧜ regularity results for Lipschitz solutions of nondegenerate quasilinear subelliptic equations of p-Laplacian type for a class of Hörmander vector fields that include certain nonnilpotent structures.
Article
We adapt a technique developed by Bojarski and Iwaniec in their celebrated 1983 paper [2] to prove second order differentiability results for $p$-harmonic functions to the case of the Heisenberg group. We prove that for $2\le p
Article
In this paper we prove second-order horizontal differentiability and C1,α regularity results for subelliptic p-harmonic functions in Carnot groups for p close to 2.
Article
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We introduce and study weighted function spaces for vector fields from the point of view of the regularity theory for quasilinear subelliptic PDEs.
Article
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In this article we propose to find the best constant for the Friedrichs-Knapp-Stein inequality in F2n,2, that is the free nilpotent Lie group of step two on 2n generators, and to prove the second-order differentiability of subelliptic p-harmonic functions in an interval of p.
Article
Full-text available
We apply subelliptic Cordes conditions and Talenti–Pucci type inequalities to prove W 2,2 and C 1,α estimates for p-harmonic functions in the Grušin plane for p near 2.
Article
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We propose to extend Talenti's estimates on the L2 norm of the second order deriva- tives of the solutions of a uniformly elliptic PDE with measurable coefficients satisfying the Cordes condition to the non-uniformly elliptic case.
Article
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We prove Cordes type estimates for subelliptic linear partial dif-ferential operators in non-divergence form with measurable coefficients in the Heisenberg group. As an application we establish interior horizontal W 2,2 -regularity for p-harmonic functions in the Heisenberg group H 1 for the range √ 17−1 2 ≤ p < 5+ √ 5 2 .
Article
We prove C1;fi regularity for p-harmonic functions in the Heisen- berg group for p in a neighborhood of 2.
Article
Full-text available
The main result of our work [Manuscr. Math. 120, No. 4, 419–433 (2006; Zbl 1185.35037)] is the C loc 1,α regularity for subelliptic p-harmonic functions in the case of the Grušin vector fields. To this goal we prove a Calderón-Zygmund inequality and an estimate for strong solutions of a linear subelliptic equation in nondivergence form with L ∞ coe...
Article
We prove Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. As an application we establish interior horizontal W 2 , 2 W^{2,2} -regularity for p-harmonic functions in the Heisenberg group H 1 {\mathbb H}^1 for the range 17 − 1 2 ≤ p > 5 + 5 2 \frac...
Article
We propose a direct method to control the first-order fractional difference quotients of solutions to quasilinear subelliptic equations in the Heisenberg group. In this way we implement iteration methods on fractional difference quotients to obtain weak differentiability in the T-direction and then second-order weak differentiability in the horizon...
Article
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The aim of the present paper is to give a new kind of point of view in the theory of variational inequalities. Our approach makes possible the study of both scalar and vector variational inequalities under a great variety of assumptions. One can include here the variational inequalities defined on reflexive or nonreflexive Banach spaces, as well as...
Article
Full-text available
In this paper our propose is to find a common term which is included in the assumptions of theorems prov- ing existence of zeros, implicit functions, fixed points or coincidence points. This new point of view allows us to weaken the assumptions which guarantee the solvability of nonlinear equations and to recommend a possible uni- fied treatment of...
Article
Full-text available
The main result of our work ((2)) is the C 1;fi loc regularity for subelliptic p-harmonic functions in the case of the Grusin vector fields. To this goal we prove a Calderon-Zygmund inequality and an estimate for strong solutions of a linear subelliptic equation in nondivergence form with L 1 coecients.
Article
New methods to the theory of parametric generalized equations governed by accretive mappings are investigated. Two implicit function theorems were proved, one for m-accretive and another for locally accretive and continuous set-valued mappings. Parametric nonlinear evolution problems were studied using the theorem for m-accretive mappings. It is sh...
Article
Full-text available
The aim of this paper is to show connections between the sensi- tivity of the solutions of variational inequalitites and properties of the solution sets of vector variational inequalities in the infinite dimen- sional setting. We will show that, in some cases, the solution set of a vector variational inequality is a bijective and continuous image o...
Article
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The purpose of this paper is to show connections between two different types of pseudomonotone mappings. The first type of pseudomonotonic-ity notion, called in this paper A-pseudomonotonicity, was introduced by S. Karamardian [8] and has been frequently used in optimization problems [3, 4, 6, 11]. The second type, called in this paper B-pseudomono...
Article
Our goal is to show that the continuity of the solutions of monotone variational inequalities with respect to perturbations is independent from the geometrical properties of reflexive Banach spaces.
Article
Our goal is to establish new methods and results in the theory of local stability of the solutions to some non-compact generalized equations, working within the realm of reflexive Banach spaces. The continuity of the projections of a fixed-point on a family of nonempty, closed, convex sets is also studied using these methods. The results included i...
Article
We prove an equivalence between implicit function theorems. The importance of this fact is that the classical implicit function theorem, its nonsmooth generalizations and results on stability of the solutions of parametric variational inequalities from the papers by A.L. Dontchev and W.W. Hager [Math. Oper. Res. 19, No. 3, 753–768 (1994; Zbl 0835.4...

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