The prescribed mean curvature equation in weakly regular domains

  • Gian Paolo Leonardi
  • Giorgio Saracco


We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a generalized Gauss–Green theorem based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a weak Young’s law for \((\Lambda ,r_{0})\)-minimizers of the perimeter.


Prescribed mean curvature Capillarity Weak normal trace Perimeter 

Mathematics Subject Classification

Primary 49K20 35J93 Secondary 49Q20 


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Authors and Affiliations

  1. 1.Dipartimento di Scienze Fisiche, Informatiche e MatematicheUniversità degli Studi di Modena e Reggio EmiliaModenaItaly
  2. 2.Department MathematikUniversität Erlangen-NürnbergErlangenGermany

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