Abstract
In this article we consider the problem to find a very weak solution \(u \in L^1(\Omega )\) of Poisson’s equation \(- \Delta u = f\) in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^N\) for a singular right hand side \(f\) under Neumann boundary conditions on \(\partial \Omega \). We prove a general existence and uniqueness theorem and discuss regularity of very weak solutions. Particularly, we are able to generalize corresponding results for Poisson’s problem with \(f \in L^1(\Omega )\) and Neumann boundary condition \(\frac{\partial u}{\partial n} = \left( -\int \nolimits _\Omega f(x)\,dx\right) \delta _{x_0}\) for a point \(x_0 \in \partial \Omega \) to non-integrable \(f\) with \(f(x)\,|x-x_0| \in L^1(\Omega )\). Applications to existence for singular data and properties of boundary Green functions are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abergel, F., Rakotoson, J.M.: Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. DCDS 33(Series A), 1809–1818 (2013)
Abe, K., Giga, Y.: Analyticity of the Stokes semigroup in spaces of bounded functions. Acta. Math. 211(1), 1–46 (2013)
Aucher, P., Qafsoui, M.: Observation on \(W^{1, p}\) estimates for divergence elliptic equations with VMO coefficients. Bull U.M.I. 5, 487–509 (2002)
Brézis, H.: Analyse fonctionnelle. Théorie et applications. Masson, Paris (1983)
Demengel, F., Demengel, G.: Espaces Fonctionnels. Utilisation dans la Résolution des Équations aux Dérivées Partielles, EDP Sciences Paris (2007)
Brézis, H., Cabré, X.: Some simple nonlinear PDE’s without solutions. Boll. Union. Math. Italy 1-B, 223–262 (1998)
Brézis, H., Cazenave, T., Martel, Y., Ramiandrisoa, A.: Blow-up for \(u_t - \Delta u = g(u)\) revisited. Adv. Differ. Equ. 1, 73–90 (1996)
Díaz, J.I., Rakotoson, J.M.: On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary. J. Funct. Anal. 257, 807–831 (2009)
Díaz, J.I., Rakotoson, J.M.: On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. DCDS Ser. A. 27, 1037–1958 (2010)
Druet, O., Robert, F.: Juncheng Wei: the Lin-Ni’s problem for mean convex domains. Memoirs of the A.M.S. 218, 1027 (2012)
Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta. Math. 172, 137–161 (1994)
Ladyzhenskaya, O.A., Uraltseva, N.: Linear and quasilinear elliptic equations. Academic Press, New York (1968)
Nečas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques (engl.: Direct Methods in the Theory of Elliptic Equation, Springer (2012)). Masson, Paris (1967)
Quittner, P., Reichel, W.: Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions. Calc. Var. 32, 429–452 (2008)
Rakotoson, J.M.: A few natural extension of the regularity of a very weak solution. Differ. Int. Equ. 24, 1125–1140 (2011)
Rakotoson, J.M.: Regularity of a very weak solution for parabolic equations and applications. Adv. Differ. Equ. 16, 867–894 (2011)
Rakotoson, J.M.: New Hardy inequalities and behaviour of linear elliptic equations. J. Funct. Anal. 263, 2893–2920 (2012)
Rakotoson, J.M.: Réarrangement Relatif: Un Instrument d’Estimations dans les Problémes aux Limites. Springer, Berlin (2008)
Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam (1987)
Roubíček, T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel (2005)
Simader, C.H.: The weak Dirichlet and Neumann problem for the Laplacian in \(L^q\) for bounded and exterior domains. Applications. In: Krbec M et al (eds) Proceedings of the Spring School held in Roudnice and Labem (1990), Nonlinear Analysis, Function Spaces and Applications vol. 4, pp. 180–223. Teubner, Leipzig (1990)
Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, New York (2007)
Ziemer, W.: Weakly Differentiable Functions. Springer, Berlin (1989)
Acknowledgments
This work has been completed when the second author was invited by the Mathematical Institute of the University of Rostock. He thanks all the staff and particularly Professor Peter Takáč for his kindness and hospitality. The research of the first author was partly supported by the German Federal Ministry of Education and Research (via DAAD Project ID 54366261). We are grateful to the anonymous referees for their valuable remarks and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Giga.
Rights and permissions
About this article
Cite this article
Merker, J., Rakotoson, JM. Very weak solutions of Poisson’s equation with singular data under Neumann boundary conditions. Calc. Var. 52, 705–726 (2015). https://doi.org/10.1007/s00526-014-0730-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-014-0730-0