Related papers: The double phase Dirichlet problem when the lowest…

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the…

The unique existence of a weak solution to the homogeneous closed Dirichlet problem on certain D-star-shaped domains is proven for a mixed elliptic-hyperbolic equation. Equations of this kind arise in models for electromagnetic wave…

We consider the inverse multiphase Stefan problem with homogeneous Dirichlet boundary condition on a bounded Lipschitz domain, where the density of the heat source is unknown in addition to the temperature and the phase transition…

Based on the need of studying the fractional boundary value problems by using variational methods, in this paper, we introduce a fundamental theory framework of fractional Sobolev space in one dimension, study the regularity of weak…

In this paper, we study a class of singular double phase problems defined on Minkowski spaces in terms of Finsler manifolds and with right-hand sides that allow a certain type of critical growth for such problems. Under very general…

In classical potential theory, one can solve the Dirichlet problem on unbounded domains such as the upper half plane. These domains have two types of boundary points; the usual finite boundary points and another point at infinity. W. Woess…

It is developed the theory of the boundary behavior of homeomorphic solutions of the Beltrami equations ${\bar{\partial}}f=\mu\,{\partial}f$ of the Sobolev class $W^{1,1}_{\rm loc}$ with respect to prime ends of domains. On this basis,…

In this paper a Pohozaev type inequality is stated for variable exponent Sobolev spaces in order to prove non existence of nontrivial weak solutions for a Dirichlet problem with non-standard growth. The obtained results generalize a…

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $\Omega$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f…

This paper is devoted to analyse the Dirichlet problem for a nonlinear elliptic equation involving the $1$--Laplacian and a total variation term, that is, the inhomogeneous case of the equation arising in the level set formulation of the…

We study self-similar solutions of a multi-phase Stefan problem for a heat equation on the half-line $x>0$ with a constant initial data and with Dirichlet or Neumann boundary conditions. In the case of Dirichlet boundary condition we prove…

In this paper, we solve the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase. A crucial ingredient is the interior $C^2$ estimate. Our result is sharp in the sense that there exist singular solutions in…

We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian type ambient spaces, and obtain general existence and uniqueness results which apply to any codimension.

In this paper we are concerned with the initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatial-temporal space. We obtain the existence of a weak…

Consider the eigenvalue problem generated by a fixed differential operator with a sign-changing weight on the eigenvalue term. We prove that as the negative part of the weight is rescaled towards negative infinity on some subregion, the…

One dimensional Stefan problems for a semi-infinite material with temperature dependent thermal coefficients are considered. Existence and uniqueness of solution are obtained imposing a Dirichlet or a Robin type condition at fixed face…

We consider the Dirichlet problem Lu = 0 in D u = g on E = boundary of D for two second order elliptic operators L_k(u) = \sum_{i,j=1}^n a_k^{ij}(x) \partial_{ij} u(x), k=0,1, in a bounded Lipschitz domain D in R^n. The coefficients…

We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the…

In this paper, we use a probabilistic approach to show that there exists a unique, bounded continuous solution to the Dirichlet boundary value problem for a general class of second order non-symmetric elliptic operators $L$ with singular…

We study the $L^p$ Dirichlet problem for the Stokes system on Lipschitz domains. For any fixed $p>2$, we show that a reverse H\"{o}lder condition with exponent $p$ is sufficient for the solvability of the Dirichlet problem with boundary…