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Bernstein and De Giorgi type problems: new results via a geometric approach

Alberto Farina, Berardino Sciunzi, Enrico Valdinoci (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form div a ( | u ( x ) | ) u ( x ) + f ( u ( x ) ) = 0 . Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in  2 and  3 and of the Bernstein problem on the flatness of minimal area graphs in  3 . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

Nguyen Ngoc Khanh (2016)

Archivum Mathematicum

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds ( M , g ) for the following general heat equation u t = Δ V u + a u log u + b u where a is a constant and b is a differentiable function defined on M × [ 0 , ) . We suppose that the Bakry-Émery curvature and the N -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently.

Liouville theorems, a priori estimates, and blow-up rates for solutions of indefinite superlinear parabolic problems

Juraj Földes (2011)

Czechoslovak Mathematical Journal

In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems.

Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations

Thomas Bartsch, Peter Poláčik, Pavol Quittner (2011)

Journal of the European Mathematical Society

We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation u t = Δ u + u p - 1 u . We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence of steady states and time-periodic solutions are also shown.

Uncertainty principles for the Weinstein transform

Hatem Mejjaoli, Makren Salhi (2011)

Czechoslovak Mathematical Journal

The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi's theorem, Beurling's theorem, and Donoho-Stark's uncertainty principle are obtained for the Weinstein transform.

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