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We investigate Lawruk elliptic boundary-value problems for homogeneous differential equations in a two-sided refined Sobolev scale. These problems contain additional unknown functions in the boundary conditions of arbitrary orders. The…

This study addresses the inverse source problem for the fractional diffusion-wave equation, characterized by a source comprising spatial and temporal components. The investigation is primarily concerned with practical scenarios where data…

In this work we establish existence and multiplicity of solutions for elliptic problem with nonlinear boundary conditions under strong resonance conditions at infinity. The nonlinearity is resonance at infinity and the reso- nance phenomena…

The periodic solutions of a type of nonlinear hyperbolic partial differential equations with a localized nonlinearity are investigated. For instance, these equations are known to describe several acoustical systems with fluid-structure…

We consider nonhomogeneous fractional $p$-Laplace equations defined on a bounded nonsmooth domain which goes beyond the Lipschitz category. Under a sufficient flatness assumption on the domain in the sense of Reifenberg, we establish…

We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis,…

We show the existence of strong solutions in Sobolev-Slobodetskii spaces to the stationary compressible Navier-Stokes equations with inflow boundary condition. Our result holds provided certain condition on the shape of the boundary around…

In this paper, we consider the following nonlinear system involving the fractional Laplacian \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. (1) \end{equation}…

This work resolves the open problem of strong singularity ($\alpha(z)> 1$) in nonlocal Kirchhoff-type equations with variable exponents through five original theorems that collectively establish a comprehensive theory. Beginning with…

This paper rediscovers a classical homogenization result for a prototypical linear elliptic boundary value problem with periodically oscillating diffusion coefficient. Unlike classical analytical approaches such as asymptotic analysis,…

In this paper we give a comprehensive treatment of a two-penalty boundary obstacle problem for a divergence form elliptic operator, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and…

The singularities that arise in elliptic boundary value problems are treated locally by a singular function boundary integral method. This method extracts the leading singular coefficients from a series expansion that describes the local…

We show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set. We make no assumptions about the open set, very mild regularity assumptions on the semilinearity, plus a coerciveness assumption…

We consider classical solutions to $-\Delta u = f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided…

This article deals with the weak and strong unique continuation principle for fractional Schr\"odinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods allow to apply the results to variable…

We consider the time-harmonic Maxwell system in a domain with a generalized impedance edge-corner, namely the presence of two generalized impedance planes that intersect at an edge. The impedance parameter can be $0, \infty$ or a finite…

The general self-adjoint elliptic boundary value problems are considered in a domain $G\subset \Bbb R^{n+1}$ with finitely many cylindrical ends. The coefficients are stabilizing (as $x\to\infty$, $x\in G$) so slowly that we can only…

A new method for numerical solving of boundary problem for ordinary differential equations with slowly varying coefficients which is aimed at better representation of solutions in the regions of their rapid oscillations or exponential…

We consider a unique continuation problem for the wave equation given data in a volumetric subset of the space time domain. In the absence of data on the lateral boundary of the space-time cylinder we prove that the solution can be…