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We study the Minkowski problem corresponding to the p-harmonic measures and obtain results previously known for harmonic measures due to Jerison (Invent Math 105(2):375–400, 1991). We show that a class of Borel measures on spheres can be prescribed by p-harmonic measures on convex domains. [ABSTRACT FROM AUTHOR]

Let n ≥ 2 and Ω be a bounded Lipschitz domain of R n . Assume that L R is a second-order divergence form elliptic operator having real-valued, bounded, symmetric, and measurable coefficients on L 2 (Ω) with the Robin boundary condition. In this article, via first obtaining the Hölder estimate of the heat kernels of L R , the authors establish a new atomic characterization of the Hardy space H L R p (Ω) associated with L R . Using this, the authors further show that, for any given p ∈ (n n + δ 0 , 1 ] , H z p (Ω) + L ∞ (Ω) = H L N p (Ω) = H L R p (Ω) ⫋ H L D p (Ω) = H r p (Ω) , where H L D p (Ω) and H L N p (Ω) denote the Hardy spaces on Ω associated with the corresponding elliptic operators respectively having the Dirichlet and the Neumann boundary conditions, H z p (Ω) and H r p (Ω) respectively denote the "supported type" and the "restricted type" Hardy spaces on Ω , and δ 0 ∈ (0 , 1 ] is the critical index depending on the operators L D , L N , and L R . The authors then obtain the boundedness of the Riesz transform ∇ L R - 1 / 2 on the Lebesgue space L p (Ω) when p ∈ (1 , ∞) [if p > 2 , some extra assumptions are needed] and its boundedness from H L R p (Ω) to L p (Ω) when p ∈ (0 , 1 ] or to H r p (Ω) when p ∈ (n n + 1 , 1 ] . As applications, the authors further obtain the global regularity estimates, in L p (Ω) when p ∈ (0 , p 0) and in H r p (Ω) when p ∈ (n n + 1 , 1 ] , for the inhomogeneous Robin problem of L R on Ω , where p 0 ∈ (2 , ∞) is a constant depending only on n, Ω , and the operator L R . The main novelties of these results are that the range (0 , p 0) of p for the global regularity estimates in the scale of L p (Ω) is sharp and that, in some sense, the space X : = H L R 1 (Ω) is also optimal to guarantee both the boundedness of ∇ L R - 1 / 2 from X to L 1 (Ω) or to H r 1 (Ω) and the global regularity estimate ‖ ∇ u ‖ L n n - 1 (Ω ; R n) ≤ C ‖ f ‖ X for inhomogeneous Robin problems with C being a positive constant independent of both u and f. [ABSTRACT FROM AUTHOR]

We will establish an ε -regularity result for weak solutions to Legendre–Hadamard elliptic systems, under the a-priori assumption that the gradient ∇ u is small in BMO. Focusing on the case of Euler–Lagrange systems to simplify the exposition, regularity results will be obtained up to the boundary, and global consequences will be explored. Extensions to general quasilinear elliptic systems and higher-order integrands is also discussed. [ABSTRACT FROM AUTHOR]

This article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator (- Δ) 2 . These fractional p -biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional p -biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties, monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces H t , p for any t ∈ R , 1 ≤ p < ∞ and s ∈ R + \ N : If u ∈ H t , p (R n) satisfies (- Δ) s u = u = 0 in a nonempty open set V, then u ≡ 0 in R n . This property of the fractional Laplacian is then used to obtain a UCP for the fractional p -biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli–Silvestre extension. [ABSTRACT FROM AUTHOR]

Let (V , μ) be an infinite, connected, locally finite weighted graph. We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality Δ u + u σ ≤ 0 in V , where Δ is the standard graph Laplacian on V and σ > 0 . For σ ∈ (0 , 1 ] , the inequality admits no nontrivial positive solution. For σ > 1 , assuming condition ( p 0 ) on (V , μ) , we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that is μ (B (o , n)) ≲ n 2 σ σ - 1 (ln n) 1 σ - 1 for some o ∈ V and all large enough n. For any ε > 0 , we can construct an example on a homogeneous tree T N with μ (B (o , n)) ≍ n 2 σ σ - 1 (ln n) 1 σ - 1 + ε , and a solution to the inequality on (T N , μ) to illustrate the sharpness of 2 σ σ - 1 and 1 σ - 1 . [ABSTRACT FROM AUTHOR]

We prove existence of a solution to the divergence equation satisfying a new Bogovski-type estimate for the difference quotients. This enables us to give an alternative proof of the interior regularity of the solution to the p-Stokes problem, completely avoiding the pressure. Moreover, as a key preliminary result we prove boundedness of Calderón–Zygmund operators with standard kernels in weighted Lebesgue and Orlicz spaces over a general domain. [ABSTRACT FROM AUTHOR]

As a complement of (Shi and Xiao in Potential Anal 45:261-298, 2016) this article addresses five more aspects of the fractional capacities relative to bounded open Lipschitz subsets of $${\mathbb {R}}^n$$ : (i) quasi-everywhere version; (ii) unique extremal; (iii) energy functional; (iv) dual representation; (v) fine behaviour of each fractional Sobolev function in $${\mathbb {R}}^n$$ . [ABSTRACT FROM AUTHOR]

The Stokes resolvent problem λ u - Δ u + ∇ ϕ = f with div (u) = 0 subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of L σ 2 (Ω) ∋ f ↦ ϕ ∈ L 2 (Ω) decays like | λ | - 1 / 2 which agrees exactly with the scaling of the equation. In comparison to that, the operator norm of this mapping under Dirichlet boundary conditions decays like | λ | - α for 0 ≤ α ≤ 1 / 4 and we show optimality of this rate, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain Ω is convex. Invoking a famous result of Grisvard (Elliptic problems in nonsmooth domains. Monographs and studies in mathematics, Pitman, 1985), we show that weak solutions u with right-hand side f ∈ L 2 (Ω ; C d) admit H 2 -regularity and further prove localized H 2 -estimates for the Stokes resolvent problem. By a generalized version of Shen's L p -extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in L p (Ω ; C d) for 2 d / (d + 2) < p < 2 d / (d - 2) (with 1 < p < ∞ if d = 2 ). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions (Shen in Arch Ration Mech Anal 205(2):395–424, 2012) on general Lipschitz domains. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation Lu=σuq+μinΩ,in the sublinear case 00. In this case u∈Lγ+q(Ω,σ)∩Lγ(Ω,μ), where γ=1 corresponds to finite energy solutions. Here Lu:=-div(A∇u) is a linear uniformly elliptic operator with bounded measurable coefficients, and σ, μ are nonnegative functions (or Radon measures), on an arbitrary domain Ω⊆Rn which possesses a positive Green function associated with L. When 0<γ≤1, this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet space W˙01,p(Ω) for 1