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The Dirichlet problem for the biharmonic equation in a Lipschitz domain

Björn E. J. Dahlberg, C. E. Kenig, G. C. Verchota (1986)

Annales de l'institut Fourier

In this paper we study and give optimal estimates for the Dirichlet problem for the biharmonic operator Δ 2 , on an arbitrary bounded Lipschitz domain D in R n . We establish existence and uniqueness results when the boundary values have first derivatives in L 2 ( ∂ D ) , and the normal derivative is in L 2 ( ∂ D ) . The resulting solution u takes the boundary values in the sense of non-tangential convergence, and the non-tangential maximal function of ∇ u is shown to be in L 2 ( ∂ D ) .

The Dirichlet problem for the degenerate Monge-Ampère equation.

Luis A. Caffarelli, Louis Nirenberg, Joel Spruck (1986)

Revista Matemática Iberoamericana

Let Ω be a bounded convex domain in Rn with smooth, strictly convex boundary ∂Ω, i.e. the principal curvatures of ∂Ω are all positive. We study the problem of finding a convex function u in Ω such that:det (uij) = 0 in Ωu = φ given on ∂Ω.

The Dirichlet problem in weighted spaces on a dihedral domain

Adam Kubica (2009)

Banach Center Publications

We examine the Dirichlet problem for the Poisson equation and the heat equation in weighted spaces of Kondrat'ev's type on a dihedral domain. The weight is a power of the distance from a distinguished axis and it depends on the order of the derivative. We also prove a priori estimates.

The Dirichlet problem with sublinear nonlinearities

Aleksandra Orpel (2002)

Annales Polonici Mathematici

We investigate the existence of solutions of the Dirichlet problem for the differential inclusion 0 ∈ Δ x ( y ) + ∂ x G ( y , x ( y ) ) for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional J ( x ) = ∫ Ω 1 / 2 | ∇ x ( y ) | ² - G ( y , x ( y ) ) d y . We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of J. We consider the case when G is subquadratic at infinity.

The discrete maximum principle for Galerkin solutions of elliptic problems

Tomáš Vejchodský (2012)

Open Mathematics

This paper provides an equivalent characterization of the discrete maximum principle for Galerkin solutions of general linear elliptic problems. The characterization is formulated in terms of the discrete Green’s function and the elliptic projection of the boundary data. This general concept is applied to the analysis of the discrete maximum principle for the higher-order finite elements in one-dimension and to the lowest-order finite elements on simplices of arbitrary dimension. The paper surveys...

The dispersion of gas exhalations and the problem of distribution of new sources on a dry hilly surface

Dien Hien Tran (1986)

Aplikace matematiky

The process of gas exhalations in the lower layer of the atmosphere and the problem of distribution of new sources of exhalations in a hilly terrain are studied. Among other, the following assumptions are introduced: (1) the terrain is a hilly one, (2) the exhalations enter a chemical reaction with the atmosphere, (3) the process is stationary, (4) the vector of wind velocity satisfies the continuity equation. The mathematical formulation of the problem then is a mixed boundary value problem for...

Currently displaying 181 – 200 of 1048