In this paper we are concerned with the existence and uniqueness of the weak solution for the weighted p-Laplacian. The purpose of this paper is to discuss in some depth the problem of solvability of Dirichlet problem, therefore all proofs are contained in some detail. The main result of the work is the existence and uniqueness of the weak solution for the Dirichlet problem provided that the weights are bounded. Furthermore, under this assumption the solution belongs to the Sobolev space $W{₀}^{1,p}\left(\Omega \right)$.

We show that the equation div $u=f$ has, in general, no Lipschitz (respectively $W^{1,1}$) solution if $f$ is $C^0$ (respectively $L^1$).

We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation. In contrast to the one-dimensional problem investigated by P. Harasim in Appl. Math. 53 (2008), No. 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness of the state solution....

This paper is devoted to the study of the weak-strong uniqueness property for full compressible magnetohydrodynamics flows. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and an additional equation which describes the evolution of the magnetic field. Using the relative entropy inequality, we prove that a weak solution coincides with the...

Let $\Omega$ be a bounded simply connected domain in the complex plane, $ℂ$. Let $N$ be a neighborhood of $\partial \Omega$, let $p$ be fixed, $1\<p\<\infty ,$ and let $\widehat{u}$ be a positive weak solution to the $p$ Laplace equation in $\Omega \cap N.$ Assume that $\widehat{u}$ has zero boundary values on $\partial \Omega$ in the Sobolev sense and extend $\widehat{u}$ to $N\setminus \Omega$ by putting $\widehat{u}\equiv 0$ on $N\setminus \Omega .$ Then there exists a positive finite Borel measure $\widehat{\mu }$ on $ℂ$ with support contained in $\partial \Omega$ and such that$$\begin{array}{c}\hfill \int |\nabla \widehat{u}{|}^{p-2}\langle \nabla \widehat{u},\nabla \phi \rangle dA=-\int \phi d\widehat{\mu }\end{array}$$whenever $\phi \in C_0^{\infty }\left(N\right).$ If $p=2$ and if $\widehat{u}$ is the Green function for $\Omega$ with pole at $x\in \Omega \setminus \overline{N}$ then the measure $\widehat{\mu }$ coincides with harmonic measure...

We consider viscosity solutions for second order differential-functional equations of parabolic type. Initial value and mixed problems are studied. Comparison theorems for subsolutions, supersolutions and solutions are considered.

We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space [...] W˙2,1p,φ(Q,ω)${\dot{W}}_{2,1}^{p,\varphi }\left(Q,\omega \right)$.

We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier’s slip boundary conditions, and on the so-called $L^{\infty }$-truncation method, used to obtain the strong convergence of the velocity...

In this paper we consider positive unbounded solutions of second order quasilinear ordinary differential equations. Our objective is to determine the asymptotic forms of unbounded solutions. An application to exterior Dirichlet problems is also given.