The method of layer potentials in Lp and endpoint spaces for elliptic operators with L∞ coefficients

By S. Hofmann, M. Mitrea, Andrew J. Morris. 2013

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We consider layer potentials associated to elliptic operators Lu=−div(A∇u) acting in the upper half‐space R+n+1 for n⩾2 , or more generally, in a Lipschitz graph domain, where the coefficient matrix A is L∞ ‐ and t ‐independent, and solutions of Lu=0 satisfy interior estimates of De Giorgi/Nash/Moser type. A ‘Calderón–Zygmund’ theory is developed for the boundedness of layer potentials, whereby sharp Lp and endpoint space bounds are deduced from L2 ‐bounds. Appropriate versions of the classical ‘jump relation’ formulae are also derived. The method of layer potentials is then used to establish well‐posedness of boundary value problems for L with data in Lp and endpoint spaces.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

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