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By an easy trick taken from caloric polynomial theory we construct a family $mathscr{B}$ of $almost regular$ domains for the caloric Dirichlet problem. $mathscr{B}$ is a basis of the Euclidean topology. This allows to build, with a basically elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain.
We show how to apply harmonic spaces potential theory in the study of the Dirichlet problem for a general class of evolution hypoelliptic partial differential equations of second order. We construct Perron-Wiener solution and we provide a sufficient
This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology,
We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = psi(x)$ where $f$ is a natural differential operator, with a restricted domain $F$, on a manifold $X$. By natural we mean operators that arise intrinsically from a given ge
We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particula
In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in $mathbb{R}^n$.