The Dirichlet Problem with Mean Curvature Operator in Minkowski Space – a Variational Approach

Abstract

In this paper we consider the Dirichlet problem with mean curvature operator in Minkowski space :where Ω ⊂ ℝ

Paths to uniqueness of critical points and applications to partial differential equations

“…where G(x, t) := t 0 g(x, s) ds on the convex set K 0 := {u ∈ W 1,∞ (Ω); |∇u| ≤ 1, u = 0 on ∂Ω}, under suitable assumptions on g, see for example [14]. Set diam(Ω) := sup{|x − y|; x, y ∈ Ω}, M := diam(Ω) 2 ,…”

“…The operator M − appears in the Born-Infeld electrostatic theory to include the principle of finiteness in Maxwell's equations; see [18,14,9]. Solutions to (1.2) must satisfy |∇u| < 1 and can be obtained by minimization of a functional in a suitable convex subset of the Sobolev space W 1,∞ (Ω).…”

Existence and multiplicity of positive solutions of a one-dimensional mean curvature equation in Minkowski space

“…By bifurcation and topological methods, the author determined the interval of parameter λ in which the above problem has zero/one/two nontrivial nonnegative solutions according to sublinear/linear/superlinear nonlinearity at zero. We refer the reader to [7][8][9][10][11][12][13][14][15] for the N -dimensional mean curvature equation in Minkowski space. In particular, for one-dimensional mean curvature equation with Dirichlet/Neumann/periodic/mixed type boundary conditions in Minkowski space, we refer the reader to [16][17][18][19][20][21][22][23][24] and the references therein.…”

Heteroclinic solutions of singular quasilinear bistable equations

“…Therefore, thanks to [5,Theorem 4.1,Corollary 4.3], see also [30,9], we know that the minimizer To obtain an uniform quantitative bound, we can proceed as in Proposition 2.11. However, we need some more restrictive assumptions on a(t), namely (a 1 ) and a ∈ C 0,1 (R) with a ′ (t) ≤ Ca(t), for a.e.…”