Overdetermined problems for the normalized 𝑝-Laplacian

In this paper, we obtain gradient continuity estimates for viscosity solutions of ∆ N p u = f in terms of the scaling critical L(n, 1) norm of f , where ∆ N p is the normalized p−Laplacian operator defined in (1.2) below. Our main result, Theorem 2.2, corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potentialĨ f q . Moreover, for f ∈ L m with m > n, we also obtain C 1,α estimates, see Theorem 2.3 below. This improves one of the regularity results in [3], where a C 1,α estimate was established depending on the L m norm of f under the additional restriction that p > 2 and m > max(2, n, p 2 ) (see Theorem 1.2 in [3]). We also mention that differently from the approach in [3], which uses methods from divergence form theory and nonlinear potential theory in the proof of Theorem 1.2, our method is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [36]. Moreover, for f continuous, our approach also gives a somewhat different proof of the C 1,α regularity result, Theorem 1.1, in [3].

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“…We now turn our attention to the relevant notion of solution to (1.1). For p ∈ R n − {0} and X = [m ij ] ∈ S(n), following [8], we define…”

Section: Notations Preliminaries and Statement Of The Main Resultsmentioning

confidence: 99%

Gradient continuity estimates for the normalized p-Poisson equation

Banerjee

Munive

2019

Commun. Contemp. Math.

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“…For symmetry problems, besides Serrin's article [15], one can consult [11], for a more comprehensive consideration of variations of the problem. Further related results can be found in [4], [6], [8], [9], [14], [16]. It should also be mentioned that there are other applied problems that lead to free boundaries, with symmetry in focus, and where the regularity of the free boundary is a priori unknown.…”

Section: Asymptotic Property (Ap)mentioning

confidence: 87%

Radial symmetry for an elliptic PDE with a free boundary

Hajj

Shahgholian

2021

Proc. Amer. Math. Soc. Ser. B

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In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u ) in B 1 , 0 ≤ u > M , in B 1 , u = M , on ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.

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“…The case when β = 0 corresponds to the Poisson problem for the normalized p−laplacian operator and this has been studied in various contexts in a number of papers. See for instance [4], [7], [9] and one can find the references therein. For general β > 0, we refer to [3] for the interior C 1,α regularity result for such equations and also to [20] and [1] for the parabolic counterpart of such results.…”

Section: Reduction To Flat Boundary Conditionsmentioning

confidence: 99%

$C^{1, α}$ Regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions

Banerjee,

Verma

2019

Preprint

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In this paper, we establish C 1,α regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior C 1,α regularity result established in [21] for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary Hölder estimates for equations which are uniformly elliptic when the gradient is either small or large.

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Overdetermined problems for the normalized 𝑝-Laplacian

Cited by 10 publications

References 19 publications

Gradient continuity estimates for the normalized p-Poisson equation

Gradient continuity estimates for the normalized p-Poisson equation

Radial symmetry for an elliptic PDE with a free boundary

$C^{1, α}$ Regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions

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