A Remark on the Analyticity of the Solutions for Non-Linear Elliptic Partial Differential Equations

Hashimoto, Yoshiaki

doi:10.3836/tjm/1170348166

Cited by 12 publications

(15 citation statements)

References 8 publications

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“…Thanks to elliptic regularity we infer that u has a smooth representative on Ω, and that the equation is satisfied in a classical sense. The analyticity of u follows from [23,Theorem 5.8.6] or [20]. Point (b) is thus proved.…”

Section: Regularity Of the Minimizersmentioning

confidence: 82%

Faber–Krahn Inequalities for the Robin-Laplacian: A Free Discontinuity Approach

Bucur

Giacomini

2015

Arch Rational Mech Anal

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We prove a full range of Faber-Krahn inequalities in a nonlinear setting and for non smooth domains, including the open case of the torsional rigidity. The key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation. As a byproduct, we obtain the best constants for a class of Poincaré inequalities with trace terms in euclidean spaces

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Section: Regularity Of the Minimizersmentioning

confidence: 82%

Faber–Krahn Inequalities for the Robin-Laplacian: A Free Discontinuity Approach

Bucur

Giacomini

2015

Arch Rational Mech Anal

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“…This means that for every h(x) ∈ L 1 (R N ) the solution w(x, z) of the elliptic equation with data w(x, 0) = h(x) will be analytic, since it is the convolution of h with P (with respect to the x variable). In fact, once we know that w ∈ C ∞ in a certain subdomain, it will be analytic by classical results on solutions of elliptic equations with analytic coefficients (see for instance [46], [33], [35]).…”

Section: Comparison Results For the Elliptic Problemmentioning

confidence: 99%

“…. Since a solution w to such equation is C ∞ in R N +1 + and the function F (x, z, u, u j , u jk ) is analytic in (z, u jj ) ∈ R + × R N +1 we can apply the main Theorem in [35] and conclude that w(x, z) is analytic in R N +1 + .…”

Section: On Symmetrizationmentioning

confidence: 99%

Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

Sire¹,

Vázquez²,

Volzone³

2017

Chin. Ann. Math. Ser. B

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We develop further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. The theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. In this paper we extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of R N with zero Dirichlet conditions outside of Ω. As an application, we derive an original proof of the corresponding fractional Faber-Krahn inequality. We also provide a more classical variational proof of the inequality.

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“…For an alternative method of proof (one fixed localization function, to the power k, and estimating in a higher order Sobolev space (instead of in L 2 ) which is also an algebra), see Kato [18] (for the equation ∆u = u 2 ) and Hashimoto [14] (for general second order non-linear analytic PDE's).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Real analyticity away from the nucleus of pseudorelativistic Hartree–Fock orbitals

et al. 2012

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We prove that the Hartree-Fock orbitals of pseudorelativistic atoms, that is, atoms where the kinetic energy of the electrons is given by the pseudorelativistic operator √ − + 1 − 1, are real analytic away from the origin. As a consequence, the quantum mechanical ground state of such atoms is never a Hartree-Fock state.Our proof is inspired by the classical proof of analyticity by nested balls of Morrey and Nirenberg. However, the technique has to be adapted to take care of the nonlocal pseudodifferential operator, the singularity of the potential at the origin, and the nonlinear terms in the equation.

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A Remark on the Analyticity of the Solutions for Non-Linear Elliptic Partial Differential Equations

Cited by 12 publications

References 8 publications

Faber–Krahn Inequalities for the Robin-Laplacian: A Free Discontinuity Approach

Faber–Krahn Inequalities for the Robin-Laplacian: A Free Discontinuity Approach

Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

Real analyticity away from the nucleus of pseudorelativistic Hartree–Fock orbitals

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