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The Dirichlet Problem for the Total Variation Flow
, 2001
"... We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we u ..."
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Cited by 37 (9 self)
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We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.
Relaxation of singular functionals defined on Sobolev spaces
 ESAIM Control Optim. Calc. Var
"... Abstract. In this paper, we consider a Borel measurable function on the space of mn matrices f:Mmn!Rtaking the value +1, such that its rankoneconvex envelope Rf is nite and satises for some xed p>1: −c0Rf(F)c(1+kFkp) for all F2Mmn; where c;c0>0. Let Ω be a given regular bounded open domain o ..."
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Cited by 13 (0 self)
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Abstract. In this paper, we consider a Borel measurable function on the space of mn matrices f:Mmn!Rtaking the value +1, such that its rankoneconvex envelope Rf is nite and satises for some xed p>1: −c0Rf(F)c(1+kFkp) for all F2Mmn; where c;c0>0. Let Ω be a given regular bounded open domain of Rn. We dene on W1;p(Ω;Rm) the functional I(u)= R Ω f(ru(x)) dx: Then, under some technical restrictions on f, we show that the relaxed functional I for the weak topology of W1;p(Ω;Rm) has the integral representation:
Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment
, 2007
"... An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and ..."
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Cited by 3 (1 self)
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An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and optimally selects the relaxation parameter in dependence on the mesh size of discretization. The finite element analysis for the relaxed problems is carried out and a numerical example is presented which confirms our analytical findings.
Some properties of the nematic radial hedgehog in the Landaude Gennes theory
 J. Math. Anal. Appl
"... In the Landaude Gennes theoretical framework of a Qtensor description of nematic liquid crystals, we consider a radial hedgehog defect with strong anchoring conditions in a ball B ⊂ R3. We show that the scalar order parameter is monotonic, and we prove uniqueness of the minimizing hedgehog below ..."
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In the Landaude Gennes theoretical framework of a Qtensor description of nematic liquid crystals, we consider a radial hedgehog defect with strong anchoring conditions in a ball B ⊂ R3. We show that the scalar order parameter is monotonic, and we prove uniqueness of the minimizing hedgehog below the spinodal temperature T ∗. 1 Introduction and
PERIODIC SOLUTIONS FOR A NONLINEAR PARABOLIC EQUATION WITH NONLINEAR BOUNDARY CONDITIONS
"... Abstract. In this paper we prove the existence of weak periodic solutions for a nonlinear parabolic equations with the Robin periodic boundary condition. The aim will be achieved by reformulating the problem in abstract form and applying some results of the maximal monotone mapping theory joint with ..."
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Abstract. In this paper we prove the existence of weak periodic solutions for a nonlinear parabolic equations with the Robin periodic boundary condition. The aim will be achieved by reformulating the problem in abstract form and applying some results of the maximal monotone mapping theory joint with the Schauder fixed point theorem. 1.
Direct approach to Lp estimates in homogenization theory
, 2006
"... Abstract: We derive interior Lpestimates for solutions of linear elliptic systems with oscillatory coefficients. The estimates are independent of ε, the small length scale of the rapid oscillations. So far, such results are based on potential theory and restricted to periodic coefficients. Our app ..."
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Abstract: We derive interior Lpestimates for solutions of linear elliptic systems with oscillatory coefficients. The estimates are independent of ε, the small length scale of the rapid oscillations. So far, such results are based on potential theory and restricted to periodic coefficients. Our approach relies on BMOestimates and an interpolation argument, gradients are treated with the help of finite differences. This allows to treat coefficients that depend on a fast and a slow variable. The estimates imply an Lpcorrector result for approximate solutions. MSC: 35B27, 49N60, 35J15 1
function that is not polyconvex M. Silhavy THEORETICAL AND APPLIED MECHANICS
"... An O(n) invariant nonnegative rank 1 convex function of linear growth is given that is not polyconvex. This answers a recent question [8, p. 182] and [5]. The polyconvex hull of the function is calculated explicitly if n = 2: 1 ..."
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An O(n) invariant nonnegative rank 1 convex function of linear growth is given that is not polyconvex. This answers a recent question [8, p. 182] and [5]. The polyconvex hull of the function is calculated explicitly if n = 2: 1
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"... ftp ejde.math.swt.edu (login: ftp) Strongly nonlinear parabolic initialboundary value problems in Orlicz spaces ∗ ..."
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ftp ejde.math.swt.edu (login: ftp) Strongly nonlinear parabolic initialboundary value problems in Orlicz spaces ∗