Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
839776 | Nonlinear Analysis: Theory, Methods & Applications | 2015 | 16 Pages |
Abstract
The variational problem of minimizing the functional uâ¦â«Î©|Du|+1pâ«Î©|Du|pââ«Î©au on a domain ΩâRn under zero boundary values, which appears for example in the theory of Bingham fluids, shows interesting phenomena such as the formation of a subset of Ω with positive measure where the minimizer is constant (a “plateau”). The corresponding Euler-Lagrange equation should be âÎ1uâÎpu=a, but the 1-Laplacian is not defined in points where Du=0. We show that it is nevertheless possible to define subsolutions and supersolutions for this equation. We show that the comparison principle is valid, and we begin developing a potential theory analogous to what is known about the p-Laplacian. In addition we apply those notions to rotationally invariant solutions, and we discuss properties of the plateau and a possible connection to the theory.
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Authors
Florian Krügel,