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Integral representation and $$L^ \infty$$ bounds for solutions of the Helmholtz equation on arbitrary open sets in $$\mathbb{R}^ 2$$ and $$\mathbb{R}^ 3$$. (English) Zbl 0824.35013
The author proves $$L^ \infty$$-bounds for functions $$u \in H^ 1_ 0(\Omega)$$, satisfying $$\Delta u \in L^ 2 (\Omega)$$, where $$\Omega$$ is an open domain in $$\mathbb{R}^ 2$$ or $$\mathbb{R}^ 3$$. The main results are the estimates $\sup_ \Omega | u |^ 2 \leq {1 \over 2 \pi} \| \Delta u \| \| \nabla u \| \text{ (in } \mathbb{R}^ 2), \quad \sup_ \Omega | u |^ 2 \leq {1 \over 2 \pi} \biggl( \| \Delta u \| \| u \| + \| \nabla u \|^ 2 \biggr) \text{ (in } \mathbb{R}^ 3).$ The proof of this result is based on several estimates for the Green’s function for the Helmholtz equation. It is shown that the constant $$1/2 \pi$$ is best possible and the functions where this constant is attained are explicitly given.

##### MSC:
 35B45 A priori estimates in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 26D10 Inequalities involving derivatives and differential and integral operators
##### Keywords:
sharp $$L^ \infty$$-bounds; best possible constant