# 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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