April  2011, 29(2): 439-451. doi: 10.3934/dcds.2011.29.439

Existence of minimizers for nonautonomous highly discontinuous scalar multiple integrals with pointwise constrained gradients

1. 

Cima-ue, Rua Romão Ramalho 59, Évora, P-7000-671, Portugal

2. 

Cima-ue, Rua Romão Ramalho 59, P-7000-671 Évora, Portugal

Received  September 2009 Revised  April 2010 Published  October 2010

We prove existence of minimizers for the multiple integral

$\int$Ω$\l(u(x),\rho_1(x,u(x)) $∇$u(x))\ \ \rho_2(x,u(x))\ dx \ \ \ on\ \ \ $W1,1u(Ω),(*)

where Ω$\subset\R^d$ is open bounded, $u:$Ω$\toR$ is in the Sobolev space u($*$)+W1,10(Ω), with boundary data $u_$$(\cdot)\in$W1,1(Ω)$\cap C^{0}$(Ω); and $\l:R$Χ$R^d\to[0,\infty]$ is superlinear $L\oxB$-measurable with $\rho_1(\cdot,\cdot),\rho_2(\cdot,\cdot)\in C^{0}($ΩΧ$R)$ both $>0$.
   One main feature of our result is the unusually weak assumption on the lagrangian: l**$(\cdot,\cdot)$ only has to be $lsc$ at $(\cdot,0)$, i.e. at zero gradient. Here l**$(s,\cdot)$ denotes the convex-closed hull of $\l(s,\cdot)$. We also treat the nonconvex case $\l(\cdot,\cdot)\ne$l**$(\cdot,\cdot)$, whenever a well-behaved relaxed minimizer is a priori known.
   Another main feature is that $\l(s,\xi)=\infty$ is freely allowed, even at zero gradient, so that (*) may be seen as the variational reformulation of optimal control problems involving implicit first-order nonsmooth scalar partial differential inclusions under state and gradient pointwise constraints.
   The general case $\int$Ω$L(x,u(x),$∇$u(x))$ is also treated, though with less natural hypotheses, but still allowing $L(x,\cdot,\xi)$ non-$lsc$ for $\xi\ne0$.

Citation: Luís Balsa Bicho, António Ornelas. Existence of minimizers for nonautonomous highly discontinuous scalar multiple integrals with pointwise constrained gradients. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 439-451. doi: 10.3934/dcds.2011.29.439
References:
[1]

L. Ambrosio, New lower semicontinuity results for integral functionals,, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 11 (1987), 1.   Google Scholar

[2]

L. B. Bicho and A. Ornelas, Existence of continuous radially monotone minimizers for convex nonautonomous multiple integrals,, preprint., ().   Google Scholar

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P. Celada, G. Cupini and M. Guidorzi, Existence and regularity of minimizers of nonconvex integrals with $p-q$ growth,, ESAIM Control Optim. Calc. Var., 13 (2007), 343.  doi: doi:10.1051/cocv:2007014.  Google Scholar

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P. Celada and S. Perrotta, Minimizing non convex, multiple integrals: A density result,, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 721.  doi: doi:10.1017/S030821050000038X.  Google Scholar

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P. Celada and S. Perrotta, On the minimum problem for nonconvex, multiple integrals of product type,, Calc. Var. Partial Differential Equations, 12 (2001), 371.  doi: doi:10.1007/PL00009918.  Google Scholar

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A. Cellina, On minima of a functional of the gradient: Necessary conditions,, Nonlinear Anal., 20 (1993), 337.  doi: doi:10.1016/0362-546X(93)90137-H.  Google Scholar

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A. Cellina, On minima of a functional of the gradient: Suficient conditions,, Nonlinear Anal., 20 (1993), 343.  doi: doi:10.1016/0362-546X(93)90138-I.  Google Scholar

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A. Cellina, On the differential inclusion $x\'\in[- 1, + 1]$,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 69 (1980), 1.   Google Scholar

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L. Cesari, "Optimization - Theory and Applications,", Springer-Verlag, (1983).   Google Scholar

[10]

B. Dacorogna, "Direct Methods in the Calculus of Variations,", 2nd edition, (2008).   Google Scholar

[11]

B. Dacorogna and P. Marcellini, "Implicit Partial Differential Equations,", Birkhäuser, (1999).   Google Scholar

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F. S. De Blasi and G. Pianigiani, On the Dirichlet problem for first order partial differential eqations. A Baire category approach,, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 13.  doi: doi:10.1007/s000300050062.  Google Scholar

[13]

F. S. De Blasi and G. Pianigiani, Baire category and boundary value problems for ordinary and partial differential inclusions under Carathéodory assumptions,, J. Differential Equations, 243 (2007), 558.  doi: doi:10.1016/j.jde.2007.05.036.  Google Scholar

[14]

V. De Cicco and G. Leoni, A chain rule in $L^1$(div $;\Omega$) and its applications to lower semicontinuity,, Calc. Var. Partial Differential Equations, 19 (2004), 23.   Google Scholar

[15]

E. DeGiorgi, G. Buttazo and G. Dal Maso, On the lower semicontinuity of certain integral functionals,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 74 (1983), 274.   Google Scholar

[16]

I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,", North-Holland, (1976).   Google Scholar

[17]

I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity,, ESAIM Control Optim. Calc. Var., 7 (2002), 69.  doi: doi:10.1051/cocv:2002004.  Google Scholar

[18]

V. Goncharov and A. Ornelas, On minima of a functional of the gradient: A continuous selection,, Nonlinear Anal., 27 (1996), 1137.  doi: doi:10.1016/0362-546X(95)00122-C.  Google Scholar

[19]

G. Friesecke, A necessary and sufficient condition for non attainment and formation of microstructure almost everywhere in scalar variational problems,, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 437.   Google Scholar

[20]

E. Giusti, "Metodi Diretti nel Calcolo delle Variazioni,", Unione Matematica Italiana, (1994).   Google Scholar

[21]

A. D. Ioffe, On lower semicontinuity of integral functionals I, II,, SIAM J. Control and Optimization, 15 (1977), 521.   Google Scholar

[22]

P. Marcellini, Non convex integrals of the calculus of variations,, in, 1446 (1990), 16.   Google Scholar

[23]

M. Marques and A. Ornelas, Genericity and existence of minimum for nonconvex scalar integral functionals,, J. Optim. Theory Appl., 86 (1995), 421.  doi: doi:10.1007/BF02192088.  Google Scholar

[24]

E. Mascolo and R. Schianchi, Existence theorems for nonconvex problems,, J. Math. Pures Appl., 62 (1983), 349.   Google Scholar

[25]

E. Mascolo and R. Schianchi, Nonconvex problems in the calculus of variations,, Nonlinear Anal., 9 (1985), 371.  doi: doi:10.1016/0362-546X(85)90060-4.  Google Scholar

[26]

A. Ornelas, Existence of scalar minimizers for simple convex integrals with autonomous Lagrangian measurable on the state variable,, Nonlinear Anal., 67 (2007), 2485.  doi: doi:10.1016/j.na.2006.08.044.  Google Scholar

[27]

R. T. Rockafellar and R. Wets, "Variational Analysis,", Springer-Verlag, (1998).  doi: doi:10.1007/978-3-642-02431-3.  Google Scholar

[28]

S. Zagatti, On the minimum problem for non convex scalar functionals,, SIAM J. Math. Anal., 37 (2005), 982.  doi: doi:10.1137/040612506.  Google Scholar

[29]

S. Zagatti, Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations,, Calc. Var. Partial Differential Equations, 31 (2008), 511.  doi: doi:10.1007/s00526-007-0124-7.  Google Scholar

show all references

References:
[1]

L. Ambrosio, New lower semicontinuity results for integral functionals,, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 11 (1987), 1.   Google Scholar

[2]

L. B. Bicho and A. Ornelas, Existence of continuous radially monotone minimizers for convex nonautonomous multiple integrals,, preprint., ().   Google Scholar

[3]

P. Celada, G. Cupini and M. Guidorzi, Existence and regularity of minimizers of nonconvex integrals with $p-q$ growth,, ESAIM Control Optim. Calc. Var., 13 (2007), 343.  doi: doi:10.1051/cocv:2007014.  Google Scholar

[4]

P. Celada and S. Perrotta, Minimizing non convex, multiple integrals: A density result,, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 721.  doi: doi:10.1017/S030821050000038X.  Google Scholar

[5]

P. Celada and S. Perrotta, On the minimum problem for nonconvex, multiple integrals of product type,, Calc. Var. Partial Differential Equations, 12 (2001), 371.  doi: doi:10.1007/PL00009918.  Google Scholar

[6]

A. Cellina, On minima of a functional of the gradient: Necessary conditions,, Nonlinear Anal., 20 (1993), 337.  doi: doi:10.1016/0362-546X(93)90137-H.  Google Scholar

[7]

A. Cellina, On minima of a functional of the gradient: Suficient conditions,, Nonlinear Anal., 20 (1993), 343.  doi: doi:10.1016/0362-546X(93)90138-I.  Google Scholar

[8]

A. Cellina, On the differential inclusion $x\'\in[- 1, + 1]$,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 69 (1980), 1.   Google Scholar

[9]

L. Cesari, "Optimization - Theory and Applications,", Springer-Verlag, (1983).   Google Scholar

[10]

B. Dacorogna, "Direct Methods in the Calculus of Variations,", 2nd edition, (2008).   Google Scholar

[11]

B. Dacorogna and P. Marcellini, "Implicit Partial Differential Equations,", Birkhäuser, (1999).   Google Scholar

[12]

F. S. De Blasi and G. Pianigiani, On the Dirichlet problem for first order partial differential eqations. A Baire category approach,, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 13.  doi: doi:10.1007/s000300050062.  Google Scholar

[13]

F. S. De Blasi and G. Pianigiani, Baire category and boundary value problems for ordinary and partial differential inclusions under Carathéodory assumptions,, J. Differential Equations, 243 (2007), 558.  doi: doi:10.1016/j.jde.2007.05.036.  Google Scholar

[14]

V. De Cicco and G. Leoni, A chain rule in $L^1$(div $;\Omega$) and its applications to lower semicontinuity,, Calc. Var. Partial Differential Equations, 19 (2004), 23.   Google Scholar

[15]

E. DeGiorgi, G. Buttazo and G. Dal Maso, On the lower semicontinuity of certain integral functionals,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 74 (1983), 274.   Google Scholar

[16]

I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,", North-Holland, (1976).   Google Scholar

[17]

I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity,, ESAIM Control Optim. Calc. Var., 7 (2002), 69.  doi: doi:10.1051/cocv:2002004.  Google Scholar

[18]

V. Goncharov and A. Ornelas, On minima of a functional of the gradient: A continuous selection,, Nonlinear Anal., 27 (1996), 1137.  doi: doi:10.1016/0362-546X(95)00122-C.  Google Scholar

[19]

G. Friesecke, A necessary and sufficient condition for non attainment and formation of microstructure almost everywhere in scalar variational problems,, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 437.   Google Scholar

[20]

E. Giusti, "Metodi Diretti nel Calcolo delle Variazioni,", Unione Matematica Italiana, (1994).   Google Scholar

[21]

A. D. Ioffe, On lower semicontinuity of integral functionals I, II,, SIAM J. Control and Optimization, 15 (1977), 521.   Google Scholar

[22]

P. Marcellini, Non convex integrals of the calculus of variations,, in, 1446 (1990), 16.   Google Scholar

[23]

M. Marques and A. Ornelas, Genericity and existence of minimum for nonconvex scalar integral functionals,, J. Optim. Theory Appl., 86 (1995), 421.  doi: doi:10.1007/BF02192088.  Google Scholar

[24]

E. Mascolo and R. Schianchi, Existence theorems for nonconvex problems,, J. Math. Pures Appl., 62 (1983), 349.   Google Scholar

[25]

E. Mascolo and R. Schianchi, Nonconvex problems in the calculus of variations,, Nonlinear Anal., 9 (1985), 371.  doi: doi:10.1016/0362-546X(85)90060-4.  Google Scholar

[26]

A. Ornelas, Existence of scalar minimizers for simple convex integrals with autonomous Lagrangian measurable on the state variable,, Nonlinear Anal., 67 (2007), 2485.  doi: doi:10.1016/j.na.2006.08.044.  Google Scholar

[27]

R. T. Rockafellar and R. Wets, "Variational Analysis,", Springer-Verlag, (1998).  doi: doi:10.1007/978-3-642-02431-3.  Google Scholar

[28]

S. Zagatti, On the minimum problem for non convex scalar functionals,, SIAM J. Math. Anal., 37 (2005), 982.  doi: doi:10.1137/040612506.  Google Scholar

[29]

S. Zagatti, Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations,, Calc. Var. Partial Differential Equations, 31 (2008), 511.  doi: doi:10.1007/s00526-007-0124-7.  Google Scholar

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