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Uniqueness results for strongly monotone operators related to Gauss measure

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Abstract

In the present paper we prove some uniqueness results for weak solutions to a class of problems, whose prototype is

$$\begin{cases}-\rm{div} & ((\varepsilon+|\triangledown{u}|^2)\frac{p-2}{2}\triangledown{u}\varphi)=f\varphi\;\;\;\;\;\rm{in}\;\;\Omega\\u=0 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \rm{on}\;\partial\Omega,\end{cases}$$

where ε ≥ 0, 1 < p < +∞, φ(x) is the density of the N-dimensional Gauss measure, Ω is an open subset of ℝN(N > 1) with Gauss measure less than one and datum f belongs to the natural dual space. When p ≤ 2 we obtain a uniqueness result for ε = 0, while for p > 2 we have to consider ε > 0 unless the sign of f is constant. Some counterexamples are given too.

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Acknowledgements

This work has been partially supported by GNAMPA of the Italian INdAM (National Institute of High Mathematics).

The second author is additionally partially founed by FFABR.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Università degli Studi di Napoli “Parthenope”, Centro Direzionale Isola, C4, 80143, Napoli, Italia

    Maria Francesca Betta & Filomena Feo

  2. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, via Cintia, 80126, Napoli, Italia

    Maria Rosaria Posteraro

Authors
  1. Maria Francesca Betta
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  2. Filomena Feo
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  3. Maria Rosaria Posteraro
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Correspondence to Filomena Feo.

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Betta, M.F., Feo, F. & Posteraro, M.R. Uniqueness results for strongly monotone operators related to Gauss measure. Isr. J. Math. 233, 297–310 (2019). https://doi.org/10.1007/s11856-019-1901-7

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  • DOI: https://doi.org/10.1007/s11856-019-1901-7

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