Trace Operator’s Range Characterization for Sobolev Spaces on Lipschitz Domains of ${ℝ}^2$

Aibèche, Aissa; Amrouche, Cherif; Bahouli, Bassem

Comptes Rendus. Mathématique, Tome 361 (2023) no. G3, pp. 587-597

We give, first, two new applications related to the range characterization of the range of trace operator in $H^2\left(\Omega \right)$. After this, we characterize the range of trace operator in the Sobolev spaces $W^{3,p}\left(\Omega \right)$ when $\Omega$ is a connected bounded domain ${ℝ}^2$ with Lipschitz-continuous boundary.

On donne, d’abord, deux nouvelles applications relatives à la caractérisation de l’image de l’opérateur trace dans $H^2\left(\Omega \right)$. Après cela, on caractérise l’image de l’opérateur trace dans les espaces de Sobolev $W^{3,p}\left(\Omega \right)$, $\Omega$ étant un domaine borné, connexe de ${ℝ}^2$ de frontière lipschitzienne.

[1] Amrouche, Cherif; Ciarlet, Philippe G.; Gratie, Liliana; Kesavan, Srinivasan On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl., Volume 86 (2006) no. 2, pp. 116-132 | DOI | MR | Zbl

[2] Amrouche, Cherif; Girault, Vivette Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czech. Math. J., Volume 44 (1994) no. 1, pp. 109-140 | DOI | MR | Zbl

[3] Amrouche, Cherif; Seloula, Nour El Houda $L^p$-theory for vector potentials and Sobolev’s inequalities for vector fields: Application to the Stokes equations with pressure boundary conditions, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 1, pp. 37-92 | DOI | MR | Zbl

[4] Buffa, Annalisa; Costabel, Martin; Sheen, Dongwoo On traces for $\mathbf{H}(\mathrm{curl},\Omega )$ for Lipschitz domains, J. Math. Anal. Appl., Volume 276 (2002) no. 2, pp. 845-867 | DOI | MR | Zbl

[5] Buffa, Annalisa; Geymonat, Giuseppe On traces for $W^{2,p}\left(\Omega \right)$ in Lipschitz domains, C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001) no. 8, pp. 699-704 | DOI | MR | Zbl

[6] Ciarlet, Philippe G.; Malin, Maria; Mardare, Cristinel On a vector version of a fundamental lemma of J. L. Lions, Chin. Ann. Math., Ser. B, Volume 39 (2018) no. 1, pp. 33-46 | DOI | MR | Zbl

[7] Dahlberg, Bjorn E. J.; Kenig, Carlos Eduardo; Verchota, Gregory C. The Dirichlet problem for the biharmonic equation in Lipschitz domain, Ann. Inst. Fourier, Volume 36 (1986) no. 3, pp. 109-135 | DOI | MR | Numdam | Zbl

[8] Durán, Ricardo G.; Muschietti, María Amelia On the traces of $W^{2,p}\left(\Omega \right)$ for a Lipschitz domain, Rev. Mat. Complut., Volume 14 (2001) no. 2, pp. 371-377 | MR | Zbl

[9] Gagliardo, Emilio Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni ni n variabili, Rend. Semin. Mat. Univ. Padova, Volume 27 (1957), pp. 284-305 | Zbl

[10] Geymonat, Giuseppe Trace theorems for Sobolev spaces on Lipschitz domains. Necessary conditions, Ann. Math. Blaise Pascal, Volume 14 (2007) no. 2, pp. 187-197 | DOI | MR | Numdam | Zbl

[11] Geymonat, Giuseppe; Krasucki, Françoise On the existence of the Airy function in Lipschitz domains. Application to the traces of $H^2\left(\Omega \right)$, C. R. Acad. Sci. Paris Sér. I Math., Volume 330 (2000) no. 5, pp. 355-360 | DOI | MR | Zbl

[12] Grisvard, Pierre Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman Advanced Publishing Program, 1985 | Zbl

[13] Maz’ya, Vladimir; Mitrea, Marius; Shaposhnikova, Tatyana O. The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients, J. Anal. Math., Volume 110 (2010), pp. 167-239 | DOI | MR | Zbl

[14] Moreau, Jean Jacques Duality characterization of strain tensor distributions in arbitrary open set, J. Math. Anal. Appl., Volume 72 (1979), pp. 760-770 | DOI | MR | Zbl

[15] Nečas, Jindřich Équations aux dérivées partielles, Séminaire de Mathématiques Supérieures, 19, Presses de l’Université de Montréal, 1966 | Zbl