Construction of planar harmonic functions.
A technique is developed for constructing the solution of in , subject to boundary conditions , on . The problem is reduced to that of finding the orthogonal projection of in onto the subspace of square integrable functions harmonic in . This problem is solved by decomposition into the closed direct (not orthogonal) sum of two subspaces for which complete orthogonal bases are known. is expressed in terms of the projections , of onto , respectively. The resulting construction...
A necessary and sufficient condition for the continuous extendibility of a solution of the Neumann problem for the Laplace equation is given.
A necessary and sufficient condition for the continuous extendibility of a solution of the third problem for the Laplace equation is given.
Let be a bounded hyperconvex domain in and set , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.
è un particolare operatore di minimizzazione per forme di Dirichlet definite su un sottoinsieme finito di un frattale che è, in un certo senso, una sorta di frontiera di . Viene talvolta chiamato mappa di rinormalizzazione ed è stato usato per definire su un analogo del funzionale e un moto Browniano. In questo lavoro si provano alcuni risultati sull'unicità dell'autoforma (rispetto a ), e sulla convergenza dell'iterata di rinormalizzata. Questi risultati sono collegati con l'unicità...
We study the boundedness in of the projections onto spaces of functions with spectrum contained in horizontal strips. We obtain some results concerning convergence along nonisotropic regions of harmonic extensions of functions in with spectrum included in these horizontal strips.