lower semicontinuous functional
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2003 ◽  
Vol 133 (6) ◽  
pp. 1361-1377 ◽  
Author(s):  
Agnieszka Kałamajska

We obtain new geometric necessary conditions for a function f to define a lower semicontinuous functional of the form If(u) = ∫Ωf(u)dx, where u satisfies a given conservation law, Pu = 0, defined by a differential operator P of degree one with constant coefficients. Those conditions imply the so-called Λ-convexity condition known as the rank-one condition when we deal with a functional of the calculus of variations. In particular, we derive some new geometric properties of quasi-convex functions and state some new questions related to the rank-one conjecture of Morrey.


2000 ◽  
Vol 160 ◽  
pp. 123-134 ◽  
Author(s):  
Haruo Nagase

AbstractIn this paper we consider the following nonlinear parabolic variational inequality; u(t) ∈ D(Φ) for all where Δp is the so-called p-Laplace operator and Φ is a proper, lower semicontinuous functional. We have obtained two results concerning to solutions of this problem. Firstly, we prove a few regularity properties of solutions. Secondly, we show the continuous dependence of solutions on given data u0 and f.


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