Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBoundary value problems for parabolic operators in a time-varying domain
207
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is possibly time varying, non-smooth, and satisfies the exterior measure condition.
rate research
Read More
We obtain the maximal regularity for the mixed Dirichlet-conormal problem in cylindrical domains with time-dependent separations, which is the first of its kind. The boundary of the domain is assumed to be Reifenberg-flat and the separation is locally sufficiently close to a Lipschitz function of $m$ variables, where $m=0,ldots,d-2$, with respect to the Hausdorff distance. We consider solutions in both $L_p$-based Sobolev spaces and $L_{q,p}$-based mixed-norm Sobolev spaces.
We study the large time behaviour of the solution of linear dispersive partial differential equations posed on a finite interval, when at least one of the prescribed boundary conditions is time periodic. We use the Q equation approach, pioneered in Fokas & Lenells 2012 and applied to linear problems on the half-line in Fokas & van der Weele 2021, to characterise necessary conditions for the solution of such problem to be periodic, at least in an asymptotic sense. We then fully describe the periodicity properties of the solution in three important illustrative examples, recovering known results for the second-order cases and establishing new results for the third order case.
This paper considers boundary value problems for a class of singular elliptic operators which appear naturally in the study of asymptotically anti-de Sitter (aAdS) spacetimes. These problems involve a singular Bessel operator acting in the normal direction. After formulating a Lopatinskii condition, elliptic estimates are established for functions supported near the boundary. The Fredholm property follows from additional hypotheses in the interior. This paper provides a rigorous framework for mode analysis on aAdS spacetimes for a wide range of boundary conditions considered in the physics literature. Completeness of eigenfunctions for some Bessel operator pencils is shown.
In this paper we consider second order parabolic partial differential equations subject to the Dirichlet boundary condition on smooth domains. We establish weighted $L_{q}$-maximal regularity in weighted Triebel-Lizorkin spaces for such parabolic problems with inhomogeneous boundary data. The weights that we consider are power weights in time and space, and yield flexibility in the optimal regularity of the initial-boundary data, allow to avoid compatibility conditions at the boundary and provide a smoothing effect. In particular, we can treat rough inhomogeneous boundary data.
In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
