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, 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), 1-42.  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), 343-358. 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), 721-741. 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), 371-398. 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), 337-341. 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), 343-347. 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), 1-6.  Google Scholar

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L. Cesari, "Optimization - Theory and Applications," Springer-Verlag, New York, 1983.  Google Scholar

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B. Dacorogna, "Direct Methods in the Calculus of Variations," 2nd edition, Springer-Verlag, Berlin, 2008.  Google Scholar

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B. Dacorogna and P. Marcellini, "Implicit Partial Differential Equations," Birkhäuser, Boston, 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), 13-34. doi: doi:10.1007/s000300050062.  Google Scholar

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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), 558-577. doi: doi:10.1016/j.jde.2007.05.036.  Google Scholar

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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), 23-51.  Google Scholar

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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), 274-282.  Google Scholar

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

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I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity, ESAIM Control Optim. Calc. Var., 7 (2002), 69-95. 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), 1137-1146. doi: doi:10.1016/0362-546X(95)00122-C.  Google Scholar

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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), 437-471.  Google Scholar

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

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A. D. Ioffe, On lower semicontinuity of integral functionals I, II, SIAM J. Control and Optimization, 15 (1977), 521-538, 991-1000.  Google Scholar

[22]

P. Marcellini, Non convex integrals of the calculus of variations, in "Methods of Nonconvex Analysis" (ed. A. Cellina), Lecture Notes in Math., 1446, Springer-Verlag, Berlin, (1990), 16-57.  Google Scholar

[23]

M. Marques and A. Ornelas, Genericity and existence of minimum for nonconvex scalar integral functionals, J. Optim. Theory Appl., 86 (1995), 421-431. doi: doi:10.1007/BF02192088.  Google Scholar

[24]

E. Mascolo and R. Schianchi, Existence theorems for nonconvex problems, J. Math. Pures Appl., 62 (1983), 349-359.  Google Scholar

[25]

E. Mascolo and R. Schianchi, Nonconvex problems in the calculus of variations, Nonlinear Anal., 9 (1985), 371-379. 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), 2485-2496. doi: doi:10.1016/j.na.2006.08.044.  Google Scholar

[27]

R. T. Rockafellar and R. Wets, "Variational Analysis," Springer-Verlag, Berlin, 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), 982-995. 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), 511-519. 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), 1-42.  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), 343-358. 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), 721-741. 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), 371-398. doi: doi:10.1007/PL00009918.  Google Scholar

[6]

A. Cellina, On minima of a functional of the gradient: Necessary conditions, Nonlinear Anal., 20 (1993), 337-341. 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), 343-347. 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), 1-6.  Google Scholar

[9]

L. Cesari, "Optimization - Theory and Applications," Springer-Verlag, New York, 1983.  Google Scholar

[10]

B. Dacorogna, "Direct Methods in the Calculus of Variations," 2nd edition, Springer-Verlag, Berlin, 2008.  Google Scholar

[11]

B. Dacorogna and P. Marcellini, "Implicit Partial Differential Equations," Birkhäuser, Boston, 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), 13-34. 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), 558-577. 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), 23-51.  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), 274-282.  Google Scholar

[16]

I. Ekeland and R. Temam, "Convex Analysis and Variational Problems," North-Holland, Amsterdam, 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), 69-95. 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), 1137-1146. 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), 437-471.  Google Scholar

[20]

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

[21]

A. D. Ioffe, On lower semicontinuity of integral functionals I, II, SIAM J. Control and Optimization, 15 (1977), 521-538, 991-1000.  Google Scholar

[22]

P. Marcellini, Non convex integrals of the calculus of variations, in "Methods of Nonconvex Analysis" (ed. A. Cellina), Lecture Notes in Math., 1446, Springer-Verlag, Berlin, (1990), 16-57.  Google Scholar

[23]

M. Marques and A. Ornelas, Genericity and existence of minimum for nonconvex scalar integral functionals, J. Optim. Theory Appl., 86 (1995), 421-431. doi: doi:10.1007/BF02192088.  Google Scholar

[24]

E. Mascolo and R. Schianchi, Existence theorems for nonconvex problems, J. Math. Pures Appl., 62 (1983), 349-359.  Google Scholar

[25]

E. Mascolo and R. Schianchi, Nonconvex problems in the calculus of variations, Nonlinear Anal., 9 (1985), 371-379. 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), 2485-2496. doi: doi:10.1016/j.na.2006.08.044.  Google Scholar

[27]

R. T. Rockafellar and R. Wets, "Variational Analysis," Springer-Verlag, Berlin, 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), 982-995. 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), 511-519. doi: doi:10.1007/s00526-007-0124-7.  Google Scholar

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