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ESTIMATES AND RIGIDITY FOR STABLE SOLUTIONS TO SOME NONLINEAR ELLIPTIC PROBLEMS

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  10 September 2020

PIETRO MIRAGLIO Open the ORCID record for PIETRO MIRAGLIO [Opens in a new window]
PIETRO MIRAGLIO*
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, via Cesare Saldini 50, 20133, Milano, Italy and Universitat Politècnica de Catalunya, Departament de Matemàtiques, Diagonal 647, 08028, Barcelona, Spain
*
e-mail: [email protected]
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Abstract

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Keywords

elliptic equationsp-Laplacianstable solutionsinequalities on hypersurfacesnonlocal operatorsDirichlet to Neumann operatorone-dimensional symmetryFourier operators

MSC classification

Primary: 35J15: Second-order elliptic equations
Secondary: 35B45: A priori estimates
Type
Abstracts of Australasian PhD Theses
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 335 - 337
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Footnotes

The thesis was written under the supervision of Enrico Valdinoci of the University of Western Australia and a significant part of the work was conducted in Australia. The degree was approved jointly by Università degli Studi di Milano (Milan, Italy) and the Universitat Politècnica de Catalunya (Barcelona, Spain) on 28 January 2020.

References

Cabré, X., ‘A new proof of the boundedness results for stable solutions to semilinear elliptic equations’, Discrete Contin. Dyn. Syst. 39 (2019), 7249–7264.10.3934/dcds.2019302CrossRefGoogle Scholar
Cabré, X., Figalli, A., Ros-Oton, X. and Serra, J., ‘Stable solutions to semilinear elliptic equations are smooth up to dimension 9’, Acta Math. 224 (2020), 187–252.10.4310/ACTA.2020.v224.n2.a1CrossRefGoogle Scholar
Cabré, X. and Miraglio, P., ‘Universal Hardy–Sobolev inequalities on hypersurfaces of Euclidean space’, Preprint, arXiv:1912.09282 (2019).Google Scholar
Cinti, E., Miraglio, P. and Valdinoci, E., ‘One-dimensional symmetry for the solutions of a three-dimensional water wave problem’, J. Geom. Anal. 30 (2020), 1804–1835.10.1007/s12220-019-00279-zCrossRefGoogle Scholar
de la Llave, R. and Valdinoci, E., ‘Symmetry for a Dirichlet–Neumann problem arising in water waves’, Math. Res. Lett. 16 (2009), 909–918.10.4310/MRL.2009.v16.n5.a13CrossRefGoogle Scholar
Dipierro, S., Miraglio, P. and Valdinoci, E., ‘Symmetry results for the solutions of a partial differential equation arising in water waves’, in: 2018 MATRIX Annals, MATRIX Book Series, 3 (Springer, Cham, 2020).Google Scholar
Miraglio, P., ‘Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian’, J. Math. Anal. Appl. 489 (2020), to appear.10.1016/j.jmaa.2020.124122CrossRefGoogle Scholar
Miraglio, P. and Valdinoci, E. ‘Energy asymptotics of a Dirichlet to Neumann problem related to water waves’, Preprint, arXiv:1909.02429 (2019).10.1088/1361-6544/ab9dcbCrossRefGoogle Scholar