Calculus of Variations and Partial Differential Equations 

About: Calculus of Variations and Partial Differential Equations is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Bounded function & Boundary (topology). It has an ISSN identifier of 0944-2669. Over the lifetime, 2844 publications have been published receiving 69338 citations. The journal is also known as: Calculus of variations (Print) & Calculus of variations and partial differential equations (Print).

Local mountain passes for semilinear elliptic problems in unbounded domains

Abstract: In [3], Floer and Weinstein consider the case N = 1 and p = 3. For a given nondegenerate critical point of the potential V, assumed globally bounded, and for 0 < E < inf V, they construct a standing wave provided that h is sufficiently small. This solution concentrates around the critical point as h --+ 0. Their method, based on an interesting Lyapunov-Schmidt reduction, was extended by Oh in [6], [7] to conclude a similar result in higher dimensions, proN+2 vided that 1 < p < g'=l" He restricts himself to potentials with "mild oscillation" at infinity, namely belonging to a Kato class. In case that V is bounded this restriction is not necessary as observed by Wang in [10].

On the existence of soliton solutions to quasilinear Schrödinger equations

Abstract: Variational techniques are applied to prove the existence of standing wave solutions for quasilinear Schrodinger equations containing strongly singular nonlinearities which include derivatives of the second order. Such equations have been derived as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Direct methods of the calculus of variations and minimax methods like the Mountain Pass Theorem are used. The difficulties introduced by the nonconvex functional \(\Phi(u)=\int | abla u|^2 u^2\) are substantially different from the semilinear case.

Contraction of convex hypersurfaces in Euclidean space

Abstract: We consider a class of fully nonlinear parabolic evolution equations for hypersurfaces in Euclidean space. A new geometrical lemma is used to prove that any strictly convex compact initial hypersurface contracts to a point in finite time, becoming spherical in shape as the limit is approached. In the particular case of the mean curvature flow this provides a simple new proof of a theorem of Huisken.

Asymptotics for the minimization of a Ginzburg-Landau functional

Abstract: LetΩ ⊂ ℝ2 be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| { abla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH g 1 ={u eH 1(Ω; ℂ);u=g on ∂Ω} whereg:∂Ω∂ → ℂ is a prescribed smooth map with ¦g¦=1 on ∂Ω∂ and deg(g, ∂Ω)=0. Let uu e be a minimizer for Ee onH g 1 . We prove that ue → u0 in $$C^{1,\alpha } (\bar \Omega )$$ as e → 0, where u0 is identified. Moreover $$\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 $$ .

Regularity for general functionals with double phase

Abstract: We prove sharp regularity results for a general class of functionals of the type $$\begin{aligned} w \mapsto \int F(x, w, Dw) \, dx, \end{aligned}$$ featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$\begin{aligned} w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx,\quad 1