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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.

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