Symmetrization in a class of first-order Hamilton-Jacobi equations. (English) Zbl 0543.35014

Giarrusso, E.; Nunziante, D.

Nonlinear Anal., Theory Methods Appl. 8, 289-299 (1984).

Isoperimetric inequalities and properties of the rearrangement of a function in the sense of Hardy and Littlewood are employed to prove comparison theorems for a class of problems of the type: \(H(Du)=f(x,u)\) in G and \(u=0\) on G under appropriate hypotheses. It is demonstrated that: \(\| u\|_{\infty}\leq \| v\|_{\infty}\) and \(\| u\|_ 1\leq \| w\|_ 1\) where u is a solution of the considered problem and v and w are the solutions spherically decreasing respectively of the problems: \(K(| Dv|)=f^*(x,0)\) in \(G^*\), \(v=0\) on \(\partial G^*\) and \(K(| Dw|)=f_*(x,0)\) in \(G^*\), \(w=0\) on \(\partial G^*\), where \(K(| y|)\leq H(y), G^*\) is the ball centered at the origin with the same measure as G, and \(f^*(.,0)\) and \(f_*(.,0)\) are respectively the decreasing and the increasing spherically symmetric rearrangement of f(.,0). An example shows that generally it is not possible to compare a.e. \(u^*\) with v or w. Only in one dimension \(u^*\leq w\).

MSC:

35F30	Boundary value problems for nonlinear first-order PDEs
35A30	Geometric theory, characteristics, transformations in context of PDEs

Keywords:

Dirichlet problem; Hamilton-Jacobi equations; symmetrization; Isoperimetric inequalities; comparison theorems; rearrangement

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