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  1. Calculus of Variations and Partial Differential Equations
  2. Calculus of Variations and Partial Differential Equations : Volume 49
  3. Calculus of Variations and Partial Differential Equations : Volume 49, Issue 3-4, March 2014
  4. Sequential weak continuity of null Lagrangians at the boundary
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Calculus of Variations and Partial Differential Equations : Volume 56
Calculus of Variations and Partial Differential Equations : Volume 55
Calculus of Variations and Partial Differential Equations : Volume 54
Calculus of Variations and Partial Differential Equations : Volume 53
Calculus of Variations and Partial Differential Equations : Volume 52
Calculus of Variations and Partial Differential Equations : Volume 51
Calculus of Variations and Partial Differential Equations : Volume 50
Calculus of Variations and Partial Differential Equations : Volume 49
Calculus of Variations and Partial Differential Equations : Volume 49, Issue 3-4, March 2014
The role of the power 3 for elliptic equations with negative exponents
Gradient bounds for anisotropic partial differential equations
On the geometry and regularity of largest subsolutions for a free boundary problem in $$\mathbf{R }^2$$ : elliptic case
Equivariant entire solutions to the elliptic system $$\Delta u=W_u(u)$$ for general $$G$$ -invariant potentials
Double tori solution to an equation of mean curvature and Newtonian potential
Vortex dynamics in the presence of excess energy for the Landau–Lifshitz–Gilbert equation
Ricci curvature and monotonicity for harmonic functions
From pointwise to local regularity for solutions of Hamilton–Jacobi equations
On the pure critical exponent problem for the $$p$$ -Laplacian
A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators
Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $${\varvec{R}}^{2n}$$
Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model
Conical singularities in thin elastic sheets
Regularity of the Monge–Ampère equation in Besov’s spaces
$$k$$ -Symplectic Pontryagin’s Maximum Principle for some families of PDEs
On the Dirichlet problem for Monge-Ampère type equations
Singular perturbation method for inhomogeneous nonlinear free boundary problems
Sequential weak continuity of null Lagrangians at the boundary
Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group
Parabolic constructions of asymptotically flat 3-metrics of prescribed scalar curvature
Geometric rigidity for sequences of $$ W^{2,2}$$ conformal immersions
Jensen’s inequality on convex spaces
The Hopf–Lax formula in Carnot groups: a control theoretic approach
Calculus of Variations and Partial Differential Equations : Volume 49, Issue 1-2, January 2014
Calculus of Variations and Partial Differential Equations : Volume 48
Calculus of Variations and Partial Differential Equations : Volume 47
Calculus of Variations and Partial Differential Equations : Volume 46
Calculus of Variations and Partial Differential Equations : Volume 45
Calculus of Variations and Partial Differential Equations : Volume 44
Calculus of Variations and Partial Differential Equations : Volume 43
Calculus of Variations and Partial Differential Equations : Volume 42
Calculus of Variations and Partial Differential Equations : Volume 41
Calculus of Variations and Partial Differential Equations : Volume 40
Calculus of Variations and Partial Differential Equations : Volume 39
Calculus of Variations and Partial Differential Equations : Volume 38
Calculus of Variations and Partial Differential Equations : Volume 37
Calculus of Variations and Partial Differential Equations : Volume 36
Calculus of Variations and Partial Differential Equations : Volume 35
Calculus of Variations and Partial Differential Equations : Volume 34
Calculus of Variations and Partial Differential Equations : Volume 33
Calculus of Variations and Partial Differential Equations : Volume 32
Calculus of Variations and Partial Differential Equations : Volume 31
Calculus of Variations and Partial Differential Equations : Volume 30
Calculus of Variations and Partial Differential Equations : Volume 29
Calculus of Variations and Partial Differential Equations : Volume 28
Calculus of Variations and Partial Differential Equations : Volume 27
Calculus of Variations and Partial Differential Equations : Volume 26
Calculus of Variations and Partial Differential Equations : Volume 25
Calculus of Variations and Partial Differential Equations : Volume 24
Calculus of Variations and Partial Differential Equations : Volume 23
Calculus of Variations and Partial Differential Equations : Volume 22
Calculus of Variations and Partial Differential Equations : Volume 21
Calculus of Variations and Partial Differential Equations : Volume 20
Calculus of Variations and Partial Differential Equations : Volume 19
Calculus of Variations and Partial Differential Equations : Volume 18
Calculus of Variations and Partial Differential Equations : Volume 17
Calculus of Variations and Partial Differential Equations : Volume 16
Calculus of Variations and Partial Differential Equations : Volume 15
Calculus of Variations and Partial Differential Equations : Volume 14
Calculus of Variations and Partial Differential Equations : Volume 13
Calculus of Variations and Partial Differential Equations : Volume 12
Calculus of Variations and Partial Differential Equations : Volume 11
Calculus of Variations and Partial Differential Equations : Volume 10
Calculus of Variations and Partial Differential Equations : Volume 9
Calculus of Variations and Partial Differential Equations : Volume 8
Calculus of Variations and Partial Differential Equations : Volume 7
Calculus of Variations and Partial Differential Equations : Volume 6
Calculus of Variations and Partial Differential Equations : Volume 5

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Sequential weak continuity of null Lagrangians at the boundary

Content Provider SpringerLink
Author Kałamajska, Agnieszka Krömer, Stefan Kružík, Martin
Copyright Year 2013
Abstract We show weak* in measures on $$\bar{\Omega }$$ / weak- $$L^1$$ sequential continuity of $$u\mapsto f(x,\nabla u):W^{1,p}(\Omega ;\mathbb{R }^m)\rightarrow L^1(\Omega )$$ , where $$f(x,\cdot )$$ is a null Lagrangian for $$x\in \Omega $$ , it is a null Lagrangian at the boundary for $$x\in \partial \Omega $$ and $$|f(x,A)|\le C(1+|A|^p)$$ . We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why $$u\mapsto \det \nabla u:W^{1,n}(\Omega ;\mathbb{R }^n)\rightarrow L^1(\Omega )$$ fails to be weakly continuous. Further, we state a new weak lower semicontinuity theorem for integrands depending on null Lagrangians at the boundary. The paper closes with an example indicating that a well-known result on higher integrability of determinant by Müller (Bull. Am. Math. Soc. New Ser. 21(2): 245–248, 1989) need not necessarily extend to our setting. The notion of quasiconvexity at the boundary due to J.M. Ball and J. Marsden is central to our analysis.
Ending Page 1278
Page Count 16
Starting Page 1263
File Format PDF
ISSN 09442669
e-ISSN 14320835
Journal Calculus of Variations and Partial Differential Equations
Issue Number 3-4
Volume Number 49
Language English
Publisher Springer Berlin Heidelberg
Publisher Date 2013-04-04
Publisher Place Berlin, Heidelberg
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Methods involving semicontinuity and convergence; relaxation Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics Oscillation, zeros of solutions, mean value theorems, etc.
Content Type Text
Resource Type Article
Subject Applied Mathematics Analysis
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