| Optimal L p estimates for the solutions of elliptic equations in bounded domains (2007) | |||||||||||||||||
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| : Linear second-order elliptic Dirichlet problems are considered in a bounded domain\Omega ae R N . Isoperimetric inequalities are used to obtain sharp estimates of the solution operator from L p to L q(\Omega\Gamma2 p; q 2 [1; 1] and 1=p ! 2=N + 1=q. The methodology used is an extension of techniques developed by Talenti. The main result removes an exponential weighting factor present in previous estimates due to Alvino, Trombetti and Lions. Keywords and phrases: Schwarz symmetrization, second-order elliptic estimates AMS(MOS) Subject classification: 35J15, 35A08, 35B45, 34B30 1 Introduction For \Omega\Gamma a bounded open subset of R N , we consider the boundary value problem L[u] j \Gamma N X i; j=1 (a ij (x)u x i ) x j \Gamma N X i=1 (b i (x)u) x i + c(x)u = f(x); x 2\Omega ; (1.1) for u zero on @ It is assumed that a ij 2 L1 (\Omega\Gamma and that, for some ? 0, N X ij=1 a ij (x) i j jj 2 ; 2 R N ; x 2\Omega : (1.2) Additionally, it is assumed that b... | |||||||||||||||||
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