ﻻ يوجد ملخص باللغة العربية
We study the second order hyperbolic equations with initial conditions, a nonhomogeneous Dirichlet boundary condition and a source term. We prove the solution possesses $H^1$ regularity on any piecewise $C^1$-smooth non-timelike hypersurfaces. We generalize the notion of energy to these hypersurfaces, and establish an estimate of the difference between the energies on the hypersurface and on the initial plane where the time $t = 0$. The energy is shown to be conserved when the source term and the boundary datum are both zero. We also obtain an $L^2$ estimate for the normal derivative of the solution. In the proofs we first show these results for $C^2$-smooth solutions by using the multiplier methods, and then we go back to the original results by approximation.
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 locall
We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a non-local equ
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-s
We consider second-order elliptic equations with oblique derivative boundary conditions, defined on a family of bounded domains in $mathbb{C}$ that depend smoothly on a real parameter $lambda in [0,1]$. We derive the precise regularity of the solutio
We present an approach to handle Dirichlet type nonlocal boundary conditions for nonlocal diffusion models with a finite range of nonlocal interactions. Our approach utilizes a linear extrapolation of prescribed boundary data. A novelty is, instead o