Related papers: Solvability of some Stefan type problems
We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary. We construct the unique self-similar solution, and show that starting from arbitrary initial data, solution orbits…
We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when…
A Stokes-type problem for a viscoelastic model of salt rocks is considered, and existence, uniqueness and regularity are investigated in the scale of $L^2$-based Sobolev spaces. The system is transformed into a generalized Stokes problem,…
We have presented a new anisotropic solution of Einstein's field equations for compact star models. The Einstein's field equations are solved by using the class one condition \cite{1}. After that we constructed the physically valid…
In this paper, we deal with the inverse problem of the shape reconstruction of inclusions in elastic bodies. The main idea of this reconstruction is based on the monotonicity property of the Neumann-to-Dirichlet operator presented in a…
We show that the free boundary of a solution of the Stefan problem in $\mathbb R^{4+1}$ is a $3$-dimensional manifold of class $C^\infty$ in $\mathbb R^4$ for almost every time. This is achieved by showing that for all dimensions $n$ the…
We consider the initial-value problem for stochastic continuity equations of the form $$ \partial_t \rho + \text{div}_h \left[\rho \left(u(t,x) + \sum_{i=1}^N a_i(x)\circ \frac{dW^i}{dt}\right)\right] = 0, $$ defined on a smooth closed…
In this article we investigate the spatial Sobolev regularity of mild solutions to stochastic Burgers equations with additive trace class noise. Our findings are based on a combination of suitable bootstrap-type arguments and a detailed…
We prove concavity properties for solutions to anisotropic quasi-linear equations, extending previous results known in the Euclidean case. We focus the attention on nonsmooth anisotropies and in particular we also allow the functions…
This paper is concerned with a class of stochastic optimization problems defined on a Banach space with almost sure conic-type constraints. For this class of problems, we investigate the consistency of optimal values and solutions…
In this paper we perform stability analysis for exponential solutions in Einstein-Gauss-Bonnet and cubic Lovelock gravity. We report our findings, provide areas on parameters space and discuss familiarities and differences between cases.…
We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point, and additional analyticity properties. This situation occurs…
We study mild solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable hyperbolicity hypotheses on the linear part. We…
We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent H\"{o}lder-continuous functions within rectangular domains. We establish a foundational approach to…
As shown in [15], under some structural assumptions, working on congested traffic problems in general and increasingly dense networks leads, at the limit by {\Gamma}-convergence, to continuous minimization problems posed on measures on…
This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem involving N-functions and theory of locally Lispchitz continuous functionals.
Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in $L^2$. In particular, we show that in relevant cases such as…
In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schr\"{o}dinger equation given by \begin{align*} \left\{ \begin{aligned} &-\epsilon^2 \Delta u+V( x)u=\lambda u+u \log u^2,…
In this paper we study the existence of solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids.…
We study the appearance of multiple solutions to certain decompositions of Einstein's constraint equations. Pfeiffer and York recently reported the existence of two branches of solutions for identical background data in the extended…