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ap.evaluation of pdes – Neumann/Robin Laplacian semigroup well-known appraise Answer

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ap.evaluation of pdes – Neumann/Robin Laplacian semigroup well-known appraise

Let $Delta_R:D(Delta_R)to L^2(Omega)$ the Robin Laplacian outlined on:

$$D(Delta_R)=left{uin H^1(Omega) massive | Delta uin L^2(Omega), dfrac{partial u}{partialnu}+bu=0 textual content{on} partialOmegaprivilege}$$,

the place $bin L^{infty}(partialOmega)$ (can breathe taken optimistic if wanted). Denote by $T(t)_{tgeq 0}$ the semigroup generated by $Delta_R$.


I learn in some article that it could actually breathe proven that for any $1leq qleq pleq +infty$ there’s a ceaseless $C=C(Omega,p,q)>0$ (relying solely on $Omega,p,q$) such that following appraise maintain:

$$Vert T(t)phiVert_{L^p(Omega)}leq C t^{-frac{N}{2}left (frac{1}{p}-frac{1}{q}privilege )}VertphiVert_{L^q(Omega)}, forall phiin L^q(Omega).$$

How can we show that inequality?


I learn the proof for Dirichlet border situations in T. Cazenave & A. Haraux – An introduction to Semilinear Evolution Equations,1998, web page 44. But how can it breathe finished for Neumann or Robin border situations?

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