Universitat des Saarlandes

UN

IVE R SIT A

S

SA

RA V I E N

SI S

Fachrichtung 6.1 Mathematik

Preprint Nr. 234

A Remark On The Global Lipschitz Regularity OfSolutions To Inner Obstacle Problems Involving

Degenerate Functionals Of p-Growth

Martin Fuchs

Saarbrucken 2009

Fachrichtung 6.1 Mathematik Preprint No. 234

Universitat des Saarlandes submitted: 20th of April, 2009

A Remark On The Global Lipschitz Regularity OfSolutions To Inner Obstacle Problems Involving

Degenerate Functionals Of p-Growth

Martin Fuchs

Saarland UniversityDep. of Mathematics

P.O. Box 15 11 50D-66041 Saarbrucken

Germanyfuchs@math.uni-sb.de

Edited byFR 6.1 MathematikUniversitat des SaarlandesPostfach 15 11 5066041 SaarbruckenGermany

Fax: + 49 681 302 4443e-Mail: preprint@math.uni-sb.deWWW: http://www.math.uni-sb.de/

AMS Subject Classification: 49 J 40, 49 N 60, 35 J 85

Keywords: inner obstacle problems, degenerate functionals, Lipschitz regularity of mini-mizers.

Abstract

We extend some recent results of Jagodzinski, Olek and Szczepaniak [JOS] on theLipschitz character of solutions to inner obstacle problems associated to a uniformlyelliptic operator to the case of nonlinear, degenerate operators.

In a recent paper Jagodzinski, Olek and Szczepaniak [JOS] investigated the Lipschitz regu-larity of solutions to socalled inner obstacle problems extending earlier work of Jordanov[Jo]. By definition we are confronted with an inner obstacle problem if one or severalside conditions imposed on the comparison functions are required to hold only on certainspecified subregions of the domain of definition, where in case of several obstacles fromabove and/or below some natural conditions for compatibility have to be satisfied. Thebasic ideas for these kind of obstacle problems with obstacles defined only on a portion of are explained in the textbook of Kinderlehrer and Stampacchia [KS] (see p.137139).The purpose of our short note now is the analysis of the inner obstacle problem for thep-energy functional

(1) I[u, ] :=1

p

|u|p dx

with arbitrary exponent p (1,). Of course we could also consider the variationalinequality associated to the operator

Lu := n

i,j=1

i (A(x,u)aij(x)ju) ,

where aij(x) are smooth elliptic coefficients and where we have abbreviated

A(x,u) := (n

,=1

a(x)uu)p

21, but this would not lead to a deeper insight.

Note that in the above cited papers the case p = 2 is considered. For simplicity we alsojust discuss the case of one inner obstacle (from below): in the presence of several innerconstraints the arguments of [JOS] have to be modified in an obvious way, which in par-ticular means that we have to impose the same natural assumptions of compatibility onthe functions acting as side conditions and on their domains of definitions as done in [JOS].

Next we give a precise formulation of our hypotheses concerning the data: let denotea bounded, open set in Rn whose boundary can locally be represented as a graph of afunction with Holder continuous derivatives. Suppose further that is an open subset of

1

with being of the same regularity as and such that . Let us consider afunction C1,1() for some 1 (0, 1) and define the class of comparison functions

(2) K := {w

W1

p() : w a.e. on } ,

where

W 1p() is the usual Sobolev space of functions vanishing on as introduced forexample in [Ad]. Then we have following result:

THEOREM 1. The inner obstacle problem I[ , ] min in K with I[ , ] and Kbeing defined in (1) and (2) admits a unique solution u K. The function u is globallyLipschitz, moreover we have u C1,( ()) for some (0, 1), () denoting theset {x : dist (x, ) < }.

Proof:Since K 6= , the existence and the uniqueness of a minimizer u is immediate, and clearlyu is the solution of the variational inequality

(3)

|u|p2u (v u) dx 0 v K .

As done in [JOS] we will reduce (3) to a global obstacle problem with a suitable constraint

: R. To this purpose consider the minimization problem

(4) I[w, ] min, w C ,

where C := {w W 1p ( ) : w| = 0 and w| = }, and let h denote the uniquesolution of (4). We further define

:=

{h on , on

}

W1

p()

and introduce the global class K := {w

W 1p() : w a.e. on } as well as theglobal problem

(5) I[ , ] min in K .

If u K denotes the unique solution of (5), we claim the validity of

(6) u = u .

In fact, u is admissible in (3), i.e. we have

(7)

|u|p2u (u u) dx 0 .

2

(5) is equivalent to the variational inequality

(8)

|u|p2u (w u) dx 0 w K .

We like to insert u into (8), which means that we have to check that

(9) u h on

holds. From (3) it follows

(10)

|u|p2u dx 0

W1

p( ), 0 ,

whereas we get from (4)

(11)

|h|p2h dx = 0

W1

p( ) .

The function := max{hu, 0} is in the space

W1p() and for this choice of (10)and (11) imply

M

[|u|p2u |h|p2h] (h u) dx 0 , M := ( ) [h > u],which by the coercivity of the field Rn 7 ||p2 immediately gives = 0 a.e.on , i.e. = 0 on this set, so that (9) follows. But then we combine (7) with (8)choosing w = u and arrive at (6).

From the works of e.g. Evans [Ev], Di Benedetto [DB], Lieberman [Li1], Manfredi [Ma1,2] and Tolksdorf [To] we deduce that the solution h of problem (4) is of class C1,()for some (0, 1), in particular there is a finite constant K such that

(12) |h| K a.e. on .

According to our assumptions || is in the space L() so that on account of (12) is Lipschitz on the whole domain . (Note that we can not guarantee the validity of

h = on , which means that we do not know if is in C1,().) This is enough toapply Theorem 1.2 of [BFM] with the result that the solution u of (5) and thereby u (recall(6)) is locally Lipschitz in . The C1,- regularity of u on the sets () is consequenceof the works of e.g. Lieberman [Li2] or Mu and Ziemer [MZ]. This completes the proof ofthe Theorem, since obviously |u| L

loc() together with u C1,( ()) implies

|u| L().

REMARK 1. If we replace the energy from (1) by a more general functional J [w, ] :=

f(w) dx, where f should at least satisfy the hypotheses of Theorem 1.2 in [BFM], then

again the global Lipschitz regularity of the minimizer will follow as soon as we can guar-antee the global boundedness of |h|, h being the solution of problem (4) now formulatedfor the functional J .

3

REMARK 2. If we assume that 0 on and if we consider the discontinuousobstacle

:=

{ on ,0 on ,

then it is easy to see that the solution u of the problem

I[ , ] min in K

coincides with the unique solution u of

I[ , ] min in K ,

K := {w

W1p() : w a.e. on } .

References

[Ad] Adams, R. A., Sobolev spaces. Academic Press, New York-San Francisco-London1975.

[BFM] Bildhauer, M., Fuchs, M., Mingione, G., A priori gradient bounds and local C1,-estimates for (double) obstacle problems under nonstandard growth conditions.Z. Anal. Anw. 20, no.4 (2001), 959985.

[DB] Di Benedetto, E., C1, local regularity results of solutions of nonlinear degenerateelliptic equations. Nonlinear Analysis, TMA, 7 (1983), 827850.

[Ev] Evans, L. C., A new proof of local C1, regularity for solutions of certain degen-erate P. D. E. s. J. Diff. Equ. 45 (1982), 356373.

[JOS] Jagodzinski, S., Olek, A., Szczepaniak, K., Lipschitz character of solutions to theinner obstacle problem. Irish Math. Soc. Bull. 61 (2008), 1527.

[Jo] Jordanov, J., Solutions holderiennes dinequations variationelles a constrainesdiscontinuous. Serdica 8 (1982), 296306.

[KS] Kinderlehrer, D., Stampacchia, G., An introduction to variational inequalitiesand their applications. Academic Press. New York-London 1980.

[Li1] Lieberman, G., Boundary regularity for solutions of degenerate elliptic equations.Nonlinear Analysis, TMA, 12 (1988), 12031219.

[Li2] Lieberman, G., Regularity of solutions to some degenerate double obstacle prob-lems. Indiana Univ. Math. J. 40 (1991), 10091028.

[Ma1] Manfredi, J., Regularity of the gradient for a class of nonlinear possibly degnerateequations. PhD thesis, Washington Univ. 1986.

4

[Ma2] Manfredi, J., Regularity for minima of functionals with p-growth. J. Diff. Equ. 76(1988), 203212.

[MZ] Mu, J., Ziemer, W. P., Smooth regularity of solutions of double obstacle problemsinvolving degenerate elliptic equations. Comm. P. D. E. 16 (485)(1991), 821843.

[To] Tolksdorf, P., Regularity for a more general class of elliptic equations.J. Diff. Equ. 51 (1984), 126150.

Martin FuchsUniversitat des SaarlandesFachbereich 6.1 MathematikPostfach 15 11 50D66041 Saarbrucken,Germanyemail: fuchs@math.uni-sb.de

5