Singular dimension of the solution set of a class of p-Laplace equations

被引:0
作者
Zubrinic, Darko [1 ]
机构
[1] Univ Zagreb, Fac Elect Engn & Comp, Dept Appl Math, Zagreb 10000, Croatia
关键词
singular set; fractal set; singular dimension; p-Laplace equation; LINEAR ELLIPTIC-EQUATIONS; GENERATING SINGULARITIES; SOBOLEV FUNCTIONS; WEAK SOLUTIONS; REGULARITY;
D O I
10.1080/17476930903568373
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We consider the boundary value problem -Delta(p)u = F(x), u is an element of W(0)(1,p) (Omega) in a bounded open domain Omega subset of R(N), where F is an element of L(p)' (Omega), 1 < p < infinity, p ' = p/(p - 1). Let X(Omega, p) be the set of weak solutions u generated by all right-hand sides F. Define the singular dimension of the solution set as the supremum of Hausdorff dimension of singular sets of solutions in X(Omega, p), and denote it by s-dim X(Omega, p). We show that for p > 2 we have s-dim X(Omega, p) (N - pp ')(+), where r(+) = max{0, r}. In the proof we exploit among others a regularity result for p-Laplace equations due to J. Simon [Sur des Equations aux Derivees Partielles Non Lineaires, These, Paris, 1977], involving Besov spaces. For 1 < p < 2, an estimate for the singular dimension of the solution set is obtained.
引用
收藏
页码:669 / 676
页数:8
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