We study existence, regularity, and positivity of solutions to linear problems involving higher-order fractional Laplacians (−∆) s for any s > 1. Using the nonlocal properties of these operators, we provide an explicit counterexample to general maximum principles for s ∈ (n, n + 1) with n ∈ N odd. In contrast, we show the validity of Boggio's representation formula for all integer and fractional powers of the Laplacian s > 0. As a consequence, maximum principles hold for weak solutions in a ball. Our proofs rely on a new variational framework based on bilinear forms, on characterizations of s-harmonic functions using higher-order Martin kernels, and on a differential recurrence equation for Boggio's formula. We also discuss the case of the whole space, where maximum principles are a consequence of the fundamental solution.

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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.

We study existence, regularity, and qualitative properties of solutions to the systemwith Ω ⊂ R N bounded; in this setting, all nontrivial solutions are sign changing. Our proofs use a variational formulation in dual spaces, considering sublinear pq < 1 and superlinear pq > 1 problems in the subcritical regime. In balls and annuli we show that least energy solutions (l.e.s.) are foliated Schwarz symmetric and, due to a symmetry-breaking phenomenon, l.e.s. are not radial functions; a key element in the proof is a new L t -norm-preserving transformation, which combines a suitable flipping with a decreasing rearrangement. This combination allows us to treat annular domains, sign-changing functions, and Neumann problems, which are non-standard settings to use rearrangements and symmetrizations. In particular, we show that our transformation diminishes the (dual) energy and, as a consequence, radial l.e.s. are strictly monotone. We also study unique continuation properties and simplicity of zeros. Our theorems also apply to the scalar associated model, where our approach provides new results as well as alternative proofs of known facts.

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfy a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e. solutions of the corresponding elliptic problem) and time periodic solutions. Mathematics Subject Classification (2010): 35B40, 35B30