67 research outputs found
The mixed problem in L^p for some two-dimensional Lipschitz domains
We consider the mixed problem for the Laplace operator in a class of
Lipschitz graph domains in two dimensions with Lipschitz constant at most 1.
The boundary of the domain is decomposed into two disjoint sets D and N. We
suppose the Dirichlet data, f_D has one derivative in L^p(D) of the boundary
and the Neumann data is in L^p(N). We find conditions on the domain and the
sets D and N so that there is a p_0>1 so that for p in the interval (1,p_0), we
may find a unique solution to the mixed problem and the gradient of the
solution lies in L^p
Bandlimited approximations to the truncated Gaussian and applications
In this paper we extend the theory of optimal approximations of functions in the -metric by entire functions of prescribed
exponential type (bandlimited functions). We solve this problem for the
truncated and the odd Gaussians using explicit integral representations and
fine properties of truncated theta functions obtained via the maximum principle
for the heat operator. As applications, we recover most of the previously known
examples in the literature and further extend the class of truncated and odd
functions for which this extremal problem can be solved, by integration on the
free parameter and the use of tempered distribution arguments. This is the
counterpart of the work \cite{CLV}, where the case of even functions is
treated.Comment: to appear in Const. Appro
Isometric Immersions and Compensated Compactness
A fundamental problem in differential geometry is to characterize intrinsic
metrics on a two-dimensional Riemannian manifold which can be
realized as isometric immersions into . This problem can be formulated as
initial and/or boundary value problems for a system of nonlinear partial
differential equations of mixed elliptic-hyperbolic type whose mathematical
theory is largely incomplete. In this paper, we develop a general approach,
which combines a fluid dynamic formulation of balance laws for the
Gauss-Codazzi system with a compensated compactness framework, to deal with the
initial and/or boundary value problems for isometric immersions in . The
compensated compactness framework formed here is a natural formulation to
ensure the weak continuity of the Gauss-Codazzi system for approximate
solutions, which yields the isometric realization of two-dimensional surfaces
in . As a first application of this approach, we study the isometric
immersion problem for two-dimensional Riemannian manifolds with strictly
negative Gauss curvature. We prove that there exists a isometric
immersion of the two-dimensional manifold in satisfying our prescribed
initial conditions. TComment: 25 pages, 6 figue
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