Related papers: On the logarithmic type boundary modulus of contin…
Let $\Omega$ be a Lipschitz domain in $\mathbb R^n$ $n\geq 2,$ and $L=\mbox{div} (A\nabla\cdot)$ be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in…
We study the elliptic equation with a line Dirac delta function as the source term subject to the Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes different types of solution singularities in the…
We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the…
We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the…
We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo's definition. We survey…
This paper aims to present a local discontinuous Galerkin (LDG) method for solving backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the $L^2$-stability and optimal error estimates of…
We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in…
We study self-similar solutions of a multi-phase Stefan problem, first in the case of one space variable, and then in the radial multidimensional case. In both these cases we prove that a nonlinear algebraic system for determination of the…
We prove existence and uniqueness of a solution for the stochastic Allen-Cahn equation with logarithmic potential and multiplicative Wiener noise, under homogeneous Neumann boundary condition. The existence of a solution is obtained in the…
We study second-order hyperbolic equations with degenerate elliptic operators and non-homogeneous Dirichlet boundary inputs. We establish existence and regularity of weak solutions in weighted Sobolev spaces under mild assumptions on the…
In this article, we prove an existence theorem regarding the weak solutions to the hyperbolic-type partial dynamic equation \begin{equation*}\begin{array}{l} z^{\Gamma\Delta}(x,y)=f(x, y, z(x, y)), z(x, 0)=0, \ \ \ z(0, y)=0 \end{array}, \…
We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragm\'en-Lindel\"of type principle…
We apply the local discontinuous Galerkin (LDG for short) method to solve a mixed boundary value problems for the Helmholtz equation in bounded polygonal domain in 2D. Under some assumptions on regularity of the solution of an adjoint…
In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional $p$-Laplacian with logarithmic nonlinearity \begin{equation*}\label{eq}\left\{ \begin{array}{llc}…
We consider a one-dimensional one-phase inverse Stefan problem for the heat equation. It consists in recovering a boundary influx condition from the knowledge of the position of the moving front, and the initial state. We derived a…
We study the continuity/discontinuity of the effective boundary condition for periodic homogenization of oscillating Dirichlet data for nonlinear divergence form equations and linear systems. For linear systems we show continuity, for…
This thesis starts from a review on current research on the local hypoellipticity of the $\bar\partial$-Neumann problem. It presents the classical method of regularity from estimates of the energy: subelliptic as well as superlogarithmic.…
We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm $L_{s,q}$-estimates hold for the spatial second-order…
By means of non-smooth critical point theory we obtain existence of infinitely many weak solutions of the fractional Schr\"odinger equation with logarithmic nonlinearity. We also investigate the H\"older regularity of the weak solutions.
In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds…