Nonlinear Partial Differential Equations and Free Boundaries: Elliptic equationsIn this Research Note the author brings together the body of known work and presents many recent results relating to nonlinear partial differential equations that give rise to a free boundary--usually the boundary of the set where the solution vanishes identically. The formation of such a boundary depends on an adequate balance between two of the terms of the equation that represent the particular characteristics of the phenomenon under consideration: diffusion, absorption, convection, evolution etc. These balances do not occur in the case of a linear equation or an arbitrary nonlinear equation. Their characterization is studied for several classes of nonlinear equations relating to applications such as chemical reactions, non-Newtonian fluids, flow through porous media and biological populations. In this first volume, the free boundary for nonlinear elliptic equations is discussed. A second volume dealing with parabolic and hyperbolic equations is in preparation. |
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Contents
THE FREE BOUNDARY IN OTHER SECOND ORDER NON LINEAR | 118 |
4a Equations in divergenece form On the diffusion | 179 |
EXISTENCE AND LOCATION OF THE FREE BOUNDARY BY MEANS | 212 |
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applied assume assumption ball Banach Banach space Benilan bounded Bp(xo BR(xo Brezis C₁ Cauchy problem Chapter coincidence set compact support comparison principle consider convex convex set defined Diaz diffusion Dirichlet problem domain elliptic equations elliptic operators estimate existence Finally free boundary F(u g and h given Hausdorff measure instance isoperimetric inequality L¹(n Lemma linear Lipschitz continuous Math maximal monotone graph maximum principle meas Moreover nondecreasing nonlinear elliptic nonlinear equations nonnegative solution Note null set N(u obstacle problem obtain operator partial differential equations proof of Theorem Proposition proved quasilinear quasilinear equations R₁ radially symmetric regularity Remark resp satisfies second order Section 1.1 semilinear equation space strong maximum principle subsection super and subsolutions supersolution Theorem 1.9 tion unique variational inequalities w¹P weak solution ΘΩ Ω Ω อน