Integral representation of solutions to higher-order fractional Dirichlet problems on balls

Abstract

We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power s > 0 of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poissontype kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if s ∈ N. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of s-harmonic functions in terms of Martin kernels, a higher-order fractional Hopf Lemma, and examples of positive and sign-changing Green functions.

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

“…Then, where E k,s are given in (1.12). A similar relationship was observed in the ball, see [4]. (v) Similar arguments to those presented for the half-space can be used to deduce the kernels for the complement of the ball B c := {x ∈ R N : |x| > 1} using the Kelvin transform (K s as in (1.17) with c = 1 and v = 0).…”

“…The Green function of (−∆) s in the unitary ball B (see [2,12]) is given by 4) where k N,s is as in (1.20).…”

“…(v) Similar arguments to those presented for the half-space can be used to deduce the kernels for the complement of the ball B c := {x ∈ R N : |x| > 1} using the Kelvin transform (K s as in (1.17) with c = 1 and v = 0). Here, similarly as in the ball case (see [4]), for k ∈ N 0 the boundary trace operator for B c is given by…”

“…Note also that, by (1.8), u can be negative even if the data of the problem is nonnegative. The assumption that g = 0 in {x ∈ R N : x 1 > −r} is needed to guarantee integrability in (1.5); this assumption can be weakened, but data which is nonzero at ∂ R N + necessarily requires a different kernel, see for example [4,Theorem 1.6] for the ball case.…”

“…The K s transformation allows us to establish a link between (1.2) and its equivalent problem on balls, for which solutions and representation formulas are known [4]. However, the case of higher-order boundary Poisson kernels is rather delicate, since K s does not map directly the trace operator of the ball to that of the half-space; in particular, these kernels require subtle and intricate combinatorial identities, which are inherent to higher-order problems.…”