- Open Access
Boundary regularity for quasilinear elliptic systems with super quadratic controllable growth condition
© Chen and Tan; licensee Springer. 2014
- Received: 3 November 2013
- Accepted: 10 April 2014
- Published: 6 May 2014
We consider the boundary regularity for weak solutions to quasilinear elliptic systems under a super quadratic controllable growth condition, and we obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on the interior partial regularity, this result yields an upper bound on the Hausdorff dimension of a singular set at the boundary.
- quasilinear elliptic systems
- controllable growth condition
- A-harmonic approximation technique
- boundary partial regularity
where Ω is a bounded domain in , , , and u and take values in . Here each maps into R, and each maps into R. For , we have the following.
for all , .
where if , or any exponent if ; for all , and .
Note that we trivially have . Further, by Sobolev’s embedding theorem we have for any . If , we will take on Ω.
If the domain we consider is an upper half unit ball , the boundary condition is the following.
Here we write , and further , . For we write for , and we set , . We further write , and we set , . For bounded with we denote the average of a given function by , i.e. . For , we set . In particular, for , , we write .
Now we can definite weak solutions to systems (1.1). Because there is a very large literature on the existence of weak solutions [1, 2], we assume that a weak solution exists  and deal with the problem of regularity directly.
for , . Obviously, the inequality remains true if we replace by , which we will henceforth abbreviate simply as .
for a constant depending only on n.
Finally, we fix an exponent as follows: if , σ can be chosen arbitrary (but henceforth fixed); otherwise we take σ fixed in .
Under such assumptions, one cannot expect that weak solutions to (1.1) will be classical . This was first shown by De Giorgi . Thus, our goal is to establish a partial regularity for weak solutions of systems (1.1).
There are some previous partial regularity results at boundary for inhomogeneous quasilinear systems. Arkhipova has studied regularity up to the boundary for nonlinear and quasilinear systems [6–8]. For systems in diagonal form, boundary regularity was first established by Wiegner , and the proof was generalized and extended by Hildebrandt-Widman . Jost-Meier  established full regularity in a neighborhood of boundary for minima of functionals with the form .
The results which are most closely related to that given here were shown in  and . In this paper, we would get the desired conclusions by the method of A-harmonic approximation. The A-harmonic approximation technique is a natural extension of harmonic approximation technique. In  Simon used harmonic approximation method to simplify Allard’s  regularity theorem and later on Schoen and Uhlenbeck’s  regularity result for harmonic maps. The idea was generalized to more general linear operators by Duzaar and Steffen , in order to deal with the regularity of almost minimizers to elliptic variational integrals in the setting of geometric measure theory. As a by-product Duzaar and Grotowski  were able to use the idea of A-harmonic approximation to deal with elliptic systems under quadratic growth, even to the boundary points for nonlinear elliptic systems  and variational problems .
In this context, we use an A-harmonic approximation method to establish boundary regularity results.
for some , , which implies .
Note in particular that the boundary condition (H5) means that makes sense: in fact, we have .
A standard covering argument  allows us to obtain the following.
Corollary 1.1 Under the assumptions of Theorem 1.1 the singular set of the weak solution u has -dimensional Hausdorff measure zero in .
If the domain of the main step in proving Theorem 1.1 is a half ball, the result then is the following.
for some , , which implies .
Analogously to above, the boundary condition (H5)′ ensures that exists, and .
Lemma 2.1 (A-harmonic approximation lemma)
Next we recall a characterization of Hölder continuous functions with a slight modification .
for all and .
for all , with the constant depending only on n and α.
We close this section by a standard estimate for the solutions to homogeneous second order elliptic systems with constant coefficients, due originally to Campanato .
In this section we prove Caccioppoli’s inequality.
Theorem 3.1 (Caccioppoli inequality)
where , , depend only on λ, L, m, and , and additionally on , , and also on s.
Proof Now we consider a cut off function , satisfying , on , and . Then the function is in , and thus it can be taken as a test-function.
here we have used the fact that and .
Fixing ε small enough yields the desired inequality immediately. □
In this section we proceed to the proof of partial regularity result.
for , .
for defined by . □
For () we have , where with . Therefore we can find such that .
We then fix , note that this also fixes δ. Since , we see from the definition of γ: , and further .
We choose small enough, such that we have .
for some , where .
for given by . We recall that this estimate is valid for all and ρ with , and we assume only the smallness condition (4.19) on . This yields after replacing R by 6R the boundary estimate required to apply Lemma 2.2.
Similarly, one can get the analogous interior estimate as (4.24). Applying Lemma 2.2, we can conclude the desired Hölder continuity. □
This article was supported by National Natural Science Foundation of China (Nos: 11201415, 11271305); Natural Science Foundation of Fujian Province (2012J01027).
- Landes R: Quasilinear elliptic operators and weak solutions of the Euler equations. Manuscr. Math. 1979, 27: 47-72.MATHMathSciNetView ArticleGoogle Scholar
- Visik IM: Quasilinear strongly elliptic systems of differential equations in divergence form. Tr. Mosk. Mat. Obŝ. 1963, 12: 140-208.MathSciNetGoogle Scholar
- Giaquinta M: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton; 1983.MATHGoogle Scholar
- Giaquinta M: A counter-example to the boundary regularity of solutions to elliptic quasilinear systems. Manuscr. Math. 1978, 24: 217-220.MATHMathSciNetView ArticleGoogle Scholar
- De Giorgi E: Frontiere orientate di misura minima. Semin. Math. Sc. Norm. Super. Pisa 1961, 57: 1-56.Google Scholar
- Arkhipova AA: Regularity results for quasilinear elliptic systems with nonlinear boundary conditions. J. Math. Sci. 1995, 77(4):3277-3294.MATHMathSciNetView ArticleGoogle Scholar
- Arkhipova AA: On the regularity of solutions of boundary-value problem for quasilinear elliptic systems with quadratic nonlinearity. J. Math. Sci. 1995, 80(6):2208-2225.MathSciNetView ArticleGoogle Scholar
- Arkhipova AA: On the regularity of the oblique derivative problem for quasilinear elliptic systems. J. Math. Sci. 1997, 84(1):817-822.MathSciNetView ArticleGoogle Scholar
- Wiegner M: A priori schranken für lösungen gewisser elliptischer systeme. Manuscr. Math. 1976, 18: 279-297.MATHMathSciNetView ArticleGoogle Scholar
- Hildebrandt S, Widman KO: On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 1977, 4: 144-178.MATHMathSciNetGoogle Scholar
- Jost J, Meier M: Boundary regularity for minima of certain quadratic functionals. Math. Ann. 1983, 262: 549-561.MATHMathSciNetView ArticleGoogle Scholar
- Grotowski JF: Boundary regularity for quasilinear elliptic systems. Commun. Partial Differ. Equ. 2002, 27(11-12):2491-2512.MATHMathSciNetView ArticleGoogle Scholar
- Grotowski JF: Boundary regularity for nonlinear elliptic systems. Calc. Var. Partial Differ. Equ. 2002, 15(3):353-388.MATHMathSciNetView ArticleGoogle Scholar
- Simon L: Lectures on Geometric Measure Theory. Australian National University Press, Canberra; 1983.MATHGoogle Scholar
- Allard WK: On the first variation of a varifold. Ann. Math. 1972, 95: 225-254.MathSciNetView ArticleGoogle Scholar
- Schoen R, Uhlenbeck K: A regularity theorem for harmonic maps. J. Differ. Geom. 1982, 17: 307-335.MATHMathSciNetGoogle Scholar
- Duzaar F, Steffen K: Optimal interior and boundary regularity for almost minimal currents to elliptic integrands. J. Reine Angew. Math. 2002, 546: 73-138.MATHMathSciNetGoogle Scholar
- Duzaar F, Grotowski JF: Partial regularity for nonlinear elliptic systems: the method of A -harmonic approximation. Manuscr. Math. 2000, 103: 267-298.MATHMathSciNetView ArticleGoogle Scholar
- Duzaar F, Kristensen J, Mingione G: The existence of regular boundary points for non-linear elliptic systems. J. Reine Angew. Math. 2007, 602: 17-58.MATHMathSciNetGoogle Scholar
- Kristensen J, Mingione G: Boundary regularity in variational problems. Arch. Ration. Mech. Anal. 2010, 198: 369-455.MATHMathSciNetView ArticleGoogle Scholar
- Campanato S: Proprietà di Hölderianità di alcune classi di funzioni. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3) 1963, 17: 175-188.MATHMathSciNetGoogle Scholar
- Campanato S:Equazioni ellittiche del ordine e spazi . Ann. Mat. Pura Appl. 1965, 69: 321-381.MATHMathSciNetView ArticleGoogle Scholar
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