Boundedness of Stable Solutions to Semilinear Elliptic Equations: A Survey

Cabré, Xavier

doi:10.1515/ans-2017-0008

Cited by 28 publications

(25 citation statements)

References 35 publications

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“…When 5 ≤ n ≤ 9, the existence of an L ∞ bound holding for all smooth nonlinearities was an open question since the mid nineties. It has been very recently solved by Figalli, Ros-Oton, Serra, and the author [8] using different ideas from those of the current article.…”

Section: Introduction and Resultsmentioning

confidence: 97%

“…This was established in 2010 by the author [5]. Since the mid nineties, the existence of an L ∞ bound holding for all smooth nonlinearities when 5 ≤ n ≤ 9 was an open problem that has been very recently solved by Figalli, Ros-Oton, Serra, and the author [8].…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…With y = 0, in the radial case this test function becomes the radial test function above. The forthcoming paper by Figalli, Ros-Oton, Serra, and the author [8] will use still a different test function to get a new key estimate.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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A new proof of the boundedness results for stable solutions to semilinear elliptic equations

Cabré¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We consider the class of stable solutions to semilinear equations −∆u = f (u) in a bounded smooth domain of R n . Since 2010 an interior a priori L ∞ bound for stable solutions is known to hold in dimensions n ≤ 4 for all C 1 nonlinearities f . In the radial case, the same is true for n ≤ 9. Here we provide with a new, simpler, and unified proof of these results. It establishes, in addition, some new estimates in higher dimensions -for instance L p bounds for every finite p in dimension 5.Since the mid nineties, the existence of an L ∞ bound holding for all C 1 nonlinearities when 5 ≤ n ≤ 9 was a challenging open problem. This has been recently solved by A. Figalli, X. Ros-Oton, J. Serra, and the author, in a forthcoming paper.

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Section: Introduction and Resultsmentioning

confidence: 97%

Section: Introduction and Resultsmentioning

confidence: 99%

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A new proof of the boundedness results for stable solutions to semilinear elliptic equations

Cabré¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…The convexity of the domain was relaxed by Villegas in [43]. Most recently, Cabré et al in [7] claimed the regularity result when n ≤ 9. For the particular nonlinearity f (u) = e u , known as the Gelfand equation, the regularity is shown u * ∈ L ∞ (Ω) for dimensions n < 10 by Crandall and Rabinowitz in [15], see also [21].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Regularity of extremal solutions of nonlocal elliptic systems

Fazly¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem    Lu = λF (u, v) in Ω, Lv = γG (u, v) in Ω,with an integro-differential operator, including the fractional Laplacian, of the formwhen J is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of J(y) = a(y/|y|) |y| n+2s where s ∈ (0, 1) and a is any nonnegative even measurable function in L 1 (S n−1 ) that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions n < 10s and n < 2s + 4s p∓1 [p + p(p ∓ 1)] for the Gelfand and Lane-Emden systems when p > 1 (with positive and negative exponents), respectively. When s → 1, these dimensions are optimal. However, for the case of s ∈ (0, 1) getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions n < 4s. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.

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“…The definition of weak solution (the sense in which u * is a solution) requires u * ∈ L 1 (Ω ), f (u * )dist(·, ∂ Ω ) ∈ L 1 (Ω ), and the equation to be satisfied in the distributional sense after multiplying it by test functions vanishing on ∂ Ω and integrating by parts twice (see [25]). Other useful references regarding extremal and stable solutions are [6], [7], and [11].…”

Section: Definition 31 (Stability)mentioning

confidence: 99%

Stable Solutions to Some Elliptic Problems: Minimal Cones, the Allen-Cahn Equation, and Blow-Up Solutions

Cabré

Poggesi

2018

Lecture Notes in Mathematics

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These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19-23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some nonlinear elliptic equations. The results are crucial steps within the regularity theory of minimizers to such problems. We focus our attention on three different equations, emphasizing that the techniques and ideas in the three settings are quite similar.The first topic is the stability of minimal cones. We prove the minimality of the Simons cone in high dimensions, and we give almost all details in the proof of J. Simons on the flatness of stable minimal cones in low dimensions.Its semilinear analogue is a conjecture on the Allen-Cahn equation posed by E. De Giorgi in 1978. This is our second problem, for which we discuss some results, as well as an open problem in high dimensions on the saddle-shaped solution vanishing on the Simons cone.The third problem was raised by H. Brezis around 1996 and concerns the boundedness of stable solutions to reaction-diffusion equations in bounded domains. We present proofs on their regularity in low dimensions and discuss the main open problem in this topic.Moreover, we briefly comment on related results for harmonic maps, free boundary problems, and nonlocal minimal surfaces. Minimal conesIn this section we discuss two classical results on the theory of minimal surfaces: Simons flatness result on stable minimal cones in low dimensions and the Bombieri-De Giorgi-Giusti counterexample in high dimensions. The main purpose of these lecture notes is to present the main ideas and computations leading to these deep results -and to related ones in subsequent sections. Therefore, to save time for this purpose, we do not consider the most general classes of sets or functions (defined through weak notions), but instead we assume them to be regular enough.Throughout the notes, for certain results we will refer to three other expositions: the books of Giusti [28] and of Colding and Minicozzi [16], and the CIME lecture notes of Cozzi and Figalli [17]. The notes [13] by the first author and Capella have a similar spirit to the current ones and may complement them. Definition 1.1 (Perimeter). Let E ⊂ R n be an open set, regular enough. For a given open ball B R we define the perimeter of E in B R aswhere H n−1 denotes the (n − 1)-dimensional Hausdorff measure (see Figure 1).The interested reader can learn from [28, 17] a more general notion of perimeter (defined by duality or in a weak sense) and the concept of set of finite perimeter. Definition 1.2 (Minimal set). We say that an open set (regular enough) E ⊂ R n is a minimal set (or a set of minimal perimeter) if and only if, for every given open ball B R , it holds that

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Boundedness of Stable Solutions to Semilinear Elliptic Equations: A Survey

Cited by 28 publications

References 35 publications

A new proof of the boundedness results for stable solutions to semilinear elliptic equations

A new proof of the boundedness results for stable solutions to semilinear elliptic equations

Regularity of extremal solutions of nonlocal elliptic systems

Stable Solutions to Some Elliptic Problems: Minimal Cones, the Allen-Cahn Equation, and Blow-Up Solutions

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