4. Variational problems with linear growth: the case of $\mu $ -elliptic integrands

2003 ◽  
pp. 97-139
Author(s):  
Michael Bildhauer
Keyword(s):  
Linear Growth ◽  
Variational Problems

scholarly journals Globally Lipschitz minimizers for variational problems with linear growth

2018 ◽  
Vol 24 (4) ◽  
pp. 1395-1413 ◽  
Author(s):  
Lisa Beck ◽  
Miroslav Bulíček ◽  
Erika Maringová
Keyword(s):  
Linear Growth ◽  
Existence Of Solutions ◽  
Direct Method ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Boundary Values ◽  
Equivalent Formulation ◽  
Radial Structure ◽  
Variational Integrals ◽  
Necessary And Sufficient

We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space W1,1 with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler–Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem. Assuming radial structure, we establish a necessary and sufficient condition on the integrand such that the Dirichlet problem is in general solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values, via the construction of appropriate barrier functions in the tradition of Serrin’s paper [J. Serrin, Philos. Trans. R. Soc. Lond., Ser. A 264 (1969) 413–496].

scholarly journals Guaranteed and computable bounds of the limit load for variational problems with linear growth energy functionals

2016 ◽  
Vol 61 (5) ◽  
pp. 527-564 ◽  
Author(s):  
Jaroslav Haslinger ◽  
Sergey Repin ◽  
Stanislav Sysala
Keyword(s):  
Linear Growth ◽  
Limit Load ◽  
Variational Problems ◽  
Energy Functionals ◽  
Growth Energy

scholarly journals On the local boundedness of generalized minimizers of variational problems with linear growth

2017 ◽  
Vol 197 (4) ◽  
pp. 1117-1129
Author(s):  
M. Bildhauer ◽  
M. Fuchs ◽  
J. Müller ◽  
X. Zhong
Keyword(s):  
Linear Growth ◽  
Variational Problems ◽  
Local Boundedness

scholarly journals On local generalized minimizers and local stress tensors for variational problems with linear growth

2010 ◽  
Vol 165 (1) ◽  
pp. 42-59 ◽  
Author(s):  
D. Apushkinskaya ◽  
M. Bildhauer ◽  
M. Fuchs
Keyword(s):  
Linear Growth ◽  
Local Stress ◽  
Variational Problems ◽  
Stress Tensors
Sign in / Sign up

Export Citation Format

Share Document