An 𝐿𝑝-Estimate for Weak Solutions of Elliptic Equations (original) (raw)

Explicit Estimates for Solutions of Mixed Elliptic Problems

International Journal of Partial Differential Equations, 2014

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in R (≥ 2) of class 0,1. The existence of ∞ and 1, estimates is assured for = 2 and any < /(− 1) (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative ∞ estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive 1, estimates for different ranges of the exponent depending on the fact that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.

A W2,p-estimate for a class of elliptic operators

International Journal of Pure and Applied Mathematics, 2013

We prove a W 2,p -a priori bound, p > 1, for a class of uniformly elliptic second order differential operators with discontinuous coefficients in unbounded domains. As an application we obtain the solvability of the related Dirichlet problem.

Domain perturbations and estimates for the solutions of second order elliptic equations

Journal de Mathématiques Pures et Appliquées, 2002

We study the dependence of the variational solution of the inhomogeneous Dirichlet problem for a second order elliptic equation with respect to perturbations of the domain. We prove optimal L 2 and energy estimates for the difference of two solutions in two open sets in terms of the "distance" between them and suitable geometrical parameters which are related to the regularity of their boundaries. We derive such estimates when at least one of the involved sets is uniformly Lipschitz: due to the connection of this problem with the regularity properties of the solutions in the L 2 family of Sobolev-Besov spaces, the Lipschitz class is the reasonably weakest one compatible with the optimal estimates.

Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form

2000

We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equation div(A(x)∇u(x)) = 0 in an n-dimensional domain , n 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain , which in turn lead to a sufficient condition for the Hölder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmén-Lindelöf type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator − div(A( · )∇u( · )) in a domain and for the solutions of the nonhomogeneous equation − div(A(x)∇u(x)) = µ with measure on the right-hand side.