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New Examples on Lavrentiev Gap Using FractalsJun 11 2019Jun 13 2019Zhikov showed 1986 with his famous checkerboard example that functionals with variable exponents can have a Lavrentiev gap. For this example it was crucial that the exponent had a saddle point whose value was exactly the dimension. In 1997 he extended ... More

New Examples on Lavrentiev Gap Using FractalsJun 11 2019Zhikov showed 1986 with his famous checkerboard example that functionals with variable exponents can have a Lavrentiev gap. For this example it was crucial that the exponent had a saddle point whose value was exactly the dimension. In 1997 he extended ... More

A unified approach to symmetry for semilinear equations associated to the Laplacian in $\mathbb{R}^N$Mar 20 2019We show radial symmetry of positive solutions to the H\'{e}non equation $-\Delta u = |x|^{-\ell} u^q $ in $\mathbb{R}^N \setminus \{ 0\} $, where $\ell \geq 0$, $q>0$ and satisfy further technical conditions. A new ingredient is a maximum principle for ... More

Min-max formulas for nonlocal elliptic operators on Euclidean spaceDec 23 2018An operator satisfies the Global Comparison Property if anytime a function touches another from above at some point, then the operator preserves the ordering at the point of contact. This is characteristic of degenerate elliptic operators, including nonlocal ... More

Nonlocal quadratic forms with visibility constraintOct 29 2018Given a subset $D$ of the Euclidean space, we study nonlocal quadratic forms that take into account tuples $(x,y) \in D \times D$ if and only if the line segment between $x$ and $y$ is contained in $D$. We discuss regularity of the corresponding Dirichlet ... More

On the Moser-Trudinger inequality in complex spaceAug 24 2018In this paper we prove the pluricomplex counterpart of the Moser-Trudinger and Sobolev inequalities in complex space. We consider these inequalities for plurisubharmonic functions with finite pluricomplex energy, and we estimate the concerned constants. ... More

Exponential decay estimates for fundamental solutions of Schrödinger-type operatorsJan 16 2018Mar 08 2019In the present paper we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators $L=-(\nabla-i\mathbf{a})^TA(\nabla-i\mathbf{a})+V$. The latter class includes, in particular, the magnetic ... More

Quadratic forms and Sobolev spaces of fractional orderJul 28 2017We study quadratic functionals on $L^2(\mathbb{R}^d)$ that generate seminorms in the fractional Sobolev space $H^s(\mathbb{R}^d)$ for $0 < s < 1$. The functionals under consideration appear in the study of Markov jump processes and, independently, in ... More

Sharp Hardy and Hardy--Sobolev inequalities with point singularities on the boundaryJan 23 2017Apr 04 2018We study the Hardy inequality when the singularity is placed on the boundary of a bounded domain in $\mathbb{R}^n$ that satisfies both an interior and exterior ball condition at the singularity. We obtain the sharp Hardy constant $n^2/4$ in case the exterior ... More

Min-max formulas for nonlocal elliptic operatorsJun 27 2016In this work, we give a characterization of Lipschitz operators on spaces of $C^2(M)$ functions (also $C^{1,1}$, $C^{1,\gamma}$, $C^1$, $C^\gamma$) that obey the global comparison property-- i.e. those that preserve the global ordering of input functions ... More

Min-max formulas for nonlocal elliptic operatorsJun 27 2016Oct 25 2016In this work, we give a characterization of Lipschitz operators on spaces of $C^2(M)$ functions (also $C^{1,1}$, $C^{1,\gamma}$, $C^1$, $C^\gamma$) that obey the global comparison property-- i.e. those that preserve the global ordering of input functions ... More

Ordinary Differential Equation Methods For Markov Decision Processes and Application to Kullback-Leibler Control CostMay 15 2016A new approach to computation of optimal policies for MDP (Markov decision process) models is introduced. The main idea is to solve not one, but an entire family of MDPs, parameterized by a weighting factor $\zeta$ that appears in the one-step reward ... More

Ordinary Differential Equation Methods For Markov Decision Processes and Application to Kullback-Leibler Control CostMay 15 2016Oct 22 2016A new approach to computation of optimal policies for MDP (Markov decision process) models is introduced. The main idea is to solve not one, but an entire family of MDPs, parameterized by a weighting factor $\zeta$ that appears in the one-step reward ... More

Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spacesMar 16 2016We study a non-homogeneous boundary value problem in a smooth bounded domain in $\mathbb{R}^N$. We prove the existence of at least two nonnegative and non-trivial weak solutions. Our approach relies on Orlicz-Sobolev spaces theory combined with adequate ... More

Nonlinear Schrödinger equations without compatibility conditions on the potentialsMay 13 2015We study the existence of nonnegative solutions (and ground states) to the nonlinear Schr\"{o}dinger equation in $\mathbb{R}^N$ with radial potentials and super-linear or sub-linear nonlinearities. The potentials satisfy power type estimates at the origin ... More

Trace Hardy--Sobolev--Mazy'a inequalities for the half fractional LaplacianSep 16 2014In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce Hardy-Sobolev-Maz'ya ... More

A family of sharp inequalities for Sobolev functionsJul 23 2014Let $N\geq 5$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, ${2^*}=\frac{2N}{N-2}$, $a>0$, $S=\inf\left\{\left. \int_{\mathbb{R}^{N}}|\nabla u|^2\,\right|\,u\in L^{2^*}(\mathbb{R}^{N}), \nabla u\in L^2(\mathbb{R}^{N}), \int_{\mathbb{R}^{N}}|u|^{2^*}=1 ... More

Compactness and existence results in weighted Sobolev spaces of radial functions, Part I: CompactnessMar 15 2014Given two measurable functions $V(r)\geq 0$ and $K(r)> 0$, $r>0$, we define the weighted spaces \[ H_V^1 = \{u \in D^{1,2}(\mathbb{R}^N): \int_{\mathbb{R}^N}V(|x|)u^{2}dx < \infty \}, \quad L_K^q = L^q(\mathbb{R}^N,K(|x|)dx) \] and study the compact embeddings ... More

Global Existence and Long-time Behaviour of Nonlinear Equation of Schrödinger typeAug 13 2012Nov 15 2012In this paper we study a mixed problem for the nonlinear Schr\"odinger equation globally that have a nonlinear adding, in which the coefficient is a generalized function. Here is proved a global solvability theorem of the considered problem with use of ... More

Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional LaplacianOct 17 2011In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy-Sobolev-Maz'ya ... More

Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean spaceJun 23 2011In this paper a quasi-linear elliptic equation in the whole Euclidean space is considered. The nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of Trudinger-Moser type inequality. Under some assumptions ... More

Dirichlet spaces on H-convex sets in Wiener spaceMay 17 2011Dec 27 2012We consider the $(1,2)$-Sobolev space $W^{1,2}(U)$ on subsets $U$ in an abstract Wiener space, which is regarded as a canonical Dirichlet space on $U$. We prove that $W^{1,2}(U)$ has smooth cylindrical functions as a dense subset if $U$ is $H$-convex ... More

Prolongation on regular infinitesimal flag manifoldsDec 08 2010Feb 24 2012Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain ... More

Resistance boundaries of infinite networksSep 08 2009Jan 22 2010A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form $\mathcal E$ produces a Hilbert space structure ${\mathcal ... More

Gel'fand triples and boundaries of infinite networksJun 15 2009Aug 16 2012We study the boundary theory of a connected weighted graph $G$ from the viewpoint of stochastic integration. For the Hilbert space \HE of Dirichlet-finite functions on $G$, we construct a Gel'fand triple $S \ci {\mathcal H}_{\mathcal E} \ci S'$. This ... More

A Hilbert space approach to effective resistance metricJun 14 2009Sep 12 2009A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form $\mathcal E$ produces a Hilbert space structure (which ... More

A discrete Gauss-Green identity for unbounded Laplace operators and transience of random walksJun 08 2009Feb 18 2010A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as resistors of possibly varying strengths. The relationship between the natural Dirichlet form $\mathcal E$ and the discrete ... More

Radial Solutions for Hamiltonian Elliptic Systems with WeightsOct 14 2008We prove the existence of infinitely many radial solutions for elliptic systems in Rn with power weights. A key tool for the proof will be a weighted imbedding theorem for fractional-order Sobolev spaces, that could be of independent interest.

Solution of the Monge-Ampere Equation on Wiener Space for log-concave measuresMar 29 2004In this work we prove that the unique 1-convex solution of the Monge problem contructed from the solution of the Monge-Kantorovitch problem between the Wiener measure and a target measure which has a log-concave density w.r.to the Wiener measure is also ... More

Lipschitz algebras and derivations II: exterior differentiationJul 17 1998Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of these cases one ... More