A Capacity-Based Condition for Existence of Solutions to Fractional Elliptic Equations with First-Order Terms and Measur
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A Capacity-Based Condition for Existence of Solutions to Fractional Elliptic Equations with First-Order Terms and Measures ´ 1 · Pablo Ochoa1 Mar´ıa Laura de Borbon Received: 2 January 2020 / Accepted: 6 August 2020 / © Springer Nature B.V. 2020
Abstract In this manuscript, we appeal to Potential Theory to provide a sufficient condition for existence of distributional solutions to fractional elliptic problems with non-linear first-order terms and measure data ω: ⎧ (−)s u = |∇u|q + ω in Rn , s ∈ (1/2, 1) ⎨ u > 0 in Rn (1) ⎩ lim|x|→∞ u(x) = 0, under suitable assumptions on q and ω. Roughly speaking, the condition for existence states that if the measure data is locally controlled by the Riesz fractional capacity, then there is a global solution for the Problem (1). We also show that if a positive solution exists, necessarily the measure ω will be absolutely continuous with respect to the associated Riesz capacity, which gives a partial reciprocal of the main result of this work. Finally, estimates of u in terms of ω are also given in different function spaces. Keywords Fractional Laplacian · Potentials and capacity · PDE’s with measures · Non-linear gradient terms Mathematics Subject Classification (2010) 35R11 · 31A15 · 35R06
1 Introduction We study the solvability of the following fractional elliptic problem ⎧ (−)s u = |∇u|q + ω in Rn ⎨ u > 0 in Rn ⎩ lim|x|→∞ u(x) = 0, Pablo Ochoa
[email protected] Mar´ıa Laura de Borb´on [email protected] 1
Universidad Nacional de Cuyo-CONICET, Mendoza 5500, Argentina
(2)
M.L. de Borb´on, P. Ochoa
where 12 < s < 1, n > 2s and (−)s is the classic fractional Laplacian operator of order 2s. Here ω will be a non-negative Radon measure with compact support in Rn . We consider the super-critical case n . q > q∗ = n − 2s + 1 This assumption on q is motivated from the fact that in the local-case, as we shall detail below, no solutions exist for sub-critical q unless ω ≡ 0. For W 1,q (Rn )-solutions, a similar conclusion is obtained in our framework (we refer the reader to Remark 1 for details). Moreover, the super-critical case allows us to obtain basic estimates on potentials as will be clarified in the proof of the main results. We highlight that non-local type operators arise naturally in continuum mechanics, image processing, crystal dislocation, Non-linear Dynamics (Geophysical Flows), phase transition phenomena, population dynamics, non-local optimal control and game theory ([5, 6, 10–12, 15, 17, 18, 20, 25] and the references therein). Indeed, models like Eq. 2 may be understood as a Kardar-Parisi-Zhang stationary problem (models of growing interfaces) driving by fractional diffusion (see [22] for the model in the local setting and [1] in the nonlocal stage). In the works [30] and [31] the description of anomalous diffusion via fractional dynamics is investigated and various fractional partial differential equations are derived from L´evy random walk models, extending Brownian motion models in a natural way. Finally, fractional type operators are also encompassed in
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