342 research outputs found
A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces
In this paper we consider scalar parabolic equations in a general non-smooth
setting with emphasis on mixed interface and boundary conditions. In
particular, we allow for dynamics and diffusion on a Lipschitz interface and on
the boundary, where diffusion coefficients are only assumed to be bounded,
measurable and positive semidefinite. In the bulk, we additionally take into
account diffusion coefficients which may degenerate towards a Lipschitz
surface. For this problem class, we introduce a unified functional analytic
framework based on sesquilinear forms and show maximal regularity for the
corresponding abstract Cauchy problem.Comment: 27 pages, 4 figure
Parabolic equations with dynamical boundary conditions and source terms on interfaces
We consider parabolic equations with mixed boundary conditions and domain
inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump
in the conormal derivative. Only minimal regularity assumptions on the domain
and the coefficients are imposed. It is shown that the corresponding linear
operator enjoys maximal parabolic regularity in a suitable -setting. The
linear results suffice to treat also the corresponding nondegenerate
quasilinear problems.Comment: 30 pages. Revised version. To appear in Annali di Matematica Pura ed
Applicat
Extrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations
In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order Xs-1,qD(Ī©) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ī© and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs
An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type
We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an -version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal -version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the -version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems
Explicit higher regularity on a Cauchy problem with mixed Neumann-power type boundary conditions
We investigate the regularity in () of the gradient of any weak
solution of a Cauchy problem with mixed Neumann-power type boundary conditions.
Under suitable assumptions we prove the existence of weak solutions that
satisfy explicit estimates. Some considerations on the steady-state regularity
are discussed.Comment: 26 page
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