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