| 1. |
- Adamowicz, Tomasz, et al.
(author)
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Regularity of p(.)-superharmonic functions, the Kellogg property and semiregular boundary points
- 2014
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In: Annales de l'Institut Henri Poincare. Analyse non linéar. - : Elsevier Masson / Institute Henri Poincar�. - 0294-1449 .- 1873-1430. ; 31:6, s. 1131-1153
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Journal article (peer-reviewed)abstract
- We study various boundary and inner regularity questions for p(.)-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for p(.)-harmonic functions into three disjoint classes: regular, semiregular and strongly irregular points. Regular and especially semiregular points are characterized in many ways. The discussion is illustrated by examples. Along the way, we present a removability result for bounded p(.)-harmonic functions and give some new characterizations of W-0(1,p(.)) spaces. We also show that p(.)-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.
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| 2. |
- Aleksanyan, Gohar, 1987-
(author)
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Optimal regularity in the optimal switching problem
- 2016
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In: Annales de l'Institut Henri Poincare. Analyse non linéar. - : Elsevier. - 0294-1449 .- 1873-1430. ; 33:6, s. 1455-1471
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Journal article (peer-reviewed)abstract
- In this article we study the optimal regularity for solutions to the following weakly coupled system with interconnected obstacles{min(−Δu1+f1,u1−u2+ψ1)=0min(−Δu2+f2,u2−u1+ψ2)=0 arising in the optimal switching problem with two modes. We derive the optimal C1,1-regularity for the minimal solution under the assumption that the zero loop set L:={ψ1+ψ2=0} is the closure of its interior. This result is optimal and we provide a counterexample showing that the C1,1-regularity does not hold without the assumption L=L0‾.
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| 3. |
- Allen, Mark, et al.
(author)
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The inhomogeneous boundary Harnack principle for fully nonlinear and p-Laplace equations
- 2023
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In: Annales de l'Institut Henri Poincare. Analyse non linéar. - : European Mathematical Society - EMS - Publishing House GmbH. - 0294-1449 .- 1873-1430. ; 40:1, s. 133-156
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Journal article (peer-reviewed)abstract
- We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and p-Laplace-type equations with a right-hand side, as well as for the Laplace equa-tion on nontangentially accessible domains under extra conditions. The approach is completely new and gives a systematic approach for proving similar results for a variety of equations and geometries.
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| 4. |
- Alvino, Angelo, et al.
(author)
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Well-posed elliptic Neumann problems involving irregular data and domains
- 2010
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In: ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE. - : GAUTHIER-VILLARS/EDITIONS ELSEVIER, 23 RUE LINOIS, 75015 PARIS, FRANCE. - 0294-1449 .- 1873-1430. ; 27:4, s. 1017-1054
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Journal article (peer-reviewed)abstract
- Non-linear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented. (C) 2010 Elsevier Masson SAS. All rights reserved.
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| 5. |
- Benedicks, Michael, et al.
(author)
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Random perturbations and statistical properties of Henon-like maps
- 2006
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In: Annales de l'Institut Henri Poincare. Analyse non linéar. - : European Mathematical Society - EMS - Publishing House GmbH. - 0294-1449 .- 1873-1430. ; 23:5, s. 713-752
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Journal article (peer-reviewed)abstract
- For a large class of non-uniformly hyperbolic diffeomorphisms, we prove stochastic stability under small random noise: the unique stationary probability measure of the Markov chain converges to the Sinai-Ruelle-Bowen measure of the unperturbed attractor when the noise level tends to zero.
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| 6. |
- Björn, Anders, et al.
(author)
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BMO, integrability, Harnack and Caccioppoli inequalities for quasiminimizers
- 2010
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In: ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE. - : GAUTHIER-VILLARS/EDITIONS ELSEVIER, 23 RUE LINOIS, 75015 PARIS, FRANCE. - 0294-1449 .- 1873-1430. ; 27:6, s. 1489-1505
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Journal article (peer-reviewed)abstract
- In this paper we use quasiminimizing properties of radial power-type functions to deduce counterexamples to certain Caccioppoli type inequalities and weak Harnack inequalities for quasisuperharmonic functions both of which are well known to hold for p-superharmonic functions We also obtain new bounds on the local integrability for quasisuperharmonic functions Furthermore we show that the logarithm of a positive quasisuperminimizer has bounded mean oscillation and belongs to a Sobolev type space
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| 7. |
- Bona, J L, et al.
(author)
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Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation
- 2005
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In: Annales de L'Institut Henri Poincaré. - : European Mathematical Society - EMS - Publishing House GmbH. - 0294-1449. ; 22:6, s. 783-797
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Journal article (peer-reviewed)abstract
- The generalized Korteweg-de Vries equation has the property that solutions with initial data that are analytic in a strip in the complex plane continue to be analytic in a strip as time progresses. Established here are algebraic lower bounds on the possible rate of decrease in time of the uniform radius of spatial analyticity for these equations. Previously known results featured exponentially decreasing bounds.
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| 8. |
- Ehrnström, Mats, et al.
(author)
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On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation
- 2019
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In: Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire. - : European Mathematical Society - EMS - Publishing House GmbH. - 0294-1449 .- 1873-1430. ; 36:6, s. 1603-1637
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Journal article (peer-reviewed)abstract
- We consider the Whitham equation u t +2uu x +Lu x =0, where L is the nonlocal Fourier multiplier operator given by the symbol m(ξ)=tanhξ/ξ. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of P-periodic solutions, and give several qualitative properties of it, including its optimal C 1/2 -regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol m(ξ), and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol m(ξ) is completely monotone, and provide an explicit representation formula for it. Our methods may be generalised.
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| 9. |
- Helsing, Johan, et al.
(author)
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Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance
- 2017
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In: Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis. - : European Mathematical Society - EMS - Publishing House GmbH. - 0294-1449 .- 1873-1430. ; 34:4, s. 991-1011
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Journal article (peer-reviewed)abstract
- We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.
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| 10. |
- Indrei, E., et al.
(author)
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Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems
- 2016
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In: Annales de l'Institut Henri Poincare. Analyse non linéar. - : Elsevier. - 0294-1449 .- 1873-1430. ; 33:5, s. 1259-1277
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Journal article (peer-reviewed)abstract
- We consider fully nonlinear obstacle-type problems of the form. F(D2u,x)=f(x)a.e. in B1∩Ω,|D2u|≤Ka.e. in B1\Ω, where Ω is an open set and K>0. In particular, structural conditions on F are presented which ensure that W2,n(B1) solutions achieve the optimal C1,1(B1/2) regularity when f is Hölder continuous. Moreover, if f is positive on B-1, Lipschitz continuous, and u≠0⊂Ω, we obtain interior C1 regularity of the free boundary under a uniform thickness assumption on u=0. Lastly, we extend these results to the parabolic setting.
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