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On the existence and uniqueness of exponentially harmonic maps and some related problems. (English) Zbl 1426.35092

In a bounded domain \(\Omega\subset \mathbb{R}^n\) the authors study the existence and uniqueness of classical solutions to the family of problems \[ \begin{cases} -\varepsilon\, \Delta u-2\,\Delta_{\infty} u=0 \quad \text{in}\quad \Omega,\\ u=g \quad \text{on}\quad \partial\Omega, \end{cases}\tag{1} \] which are equivalent to the problems \[ \begin{cases} -\text{div}\big(e^{\varepsilon^{-1}\,|\nabla u|^2}\,\nabla u\big)=0 \quad \text{in}\quad \Omega,\\ u=g \quad \text{on}\quad \partial\Omega. \end{cases} \] It is proved that the classical solution to the problem (1) is the unique minimizer element \(u_{\varepsilon}\) of the Euler-Lagrange functional \(J_{\varepsilon}(u)=\int_{\Omega}\,e^{\varepsilon^{-1}\,|\nabla u|}\,dx\) in a closed subset of an Orlicz-Sobolev space. Also the authors study the behavior of the solutions \(u_{\varepsilon}\) as \(\varepsilon\to 0^{+}\) and \(\varepsilon \to \infty\), respectively.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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