On Dirichlet's principle and Poincare's methode de balayage

Math. Nachr. 278, No. 1–2, 141 – 144 (2005) / DOI 10.1002/mana.200310231

On Dirichlet’s principle and Poincar´e’s m´ethode de balayage Stefan Hildebrandt∗1 1

Mathematisches Institut der Universit¨at Bonn, Beringstrasse 1, 53115 Bonn, Germany Received 18 September 2003, accepted 18 June 2004 Published online 15 December 2004 Key words Dirichlet’s principle, harmonic functions, boundary behaviour, regular boundary points MSC (2000) 35J05, 35J20, 35J25, 35J67 A short proof is given for the fact that the minimizer of Dirichlet’s integral continuously assumes continuous boundary values at regular boundary points. c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


The following result is an optimal version of Dirichlet’s principle: Theorem 1 Let Ω be a
bounded domain in Rn , and denote by Σ the set of its regular boundary points. Then,  1 0 for any g ∈ H (Ω) ∩ C Ω , the minimizer u of Dirichlet’s integral D(v) :=

1 2

 Ω

|∇v|2 dx

in the set   ◦ 1 1 C(Ω, g) := v ∈ H (Ω) : v − g ∈ H (Ω)
 is continuous on Ω ∪ Σ and assumes the boundary values g on Σ. In particular u ∈ C 0 Ω if all points of ∂Ω are regular. A proof of this result follows from Theorem (9.2) in Littman–Stampacchia–Weinberger [5] together with Frostman’s Wiener criterion in [2]; see also Frehse [1], §3. Recently, C. Simader [6] has kindly communicated to me an interesting “elementary” proof. It is the purpose of the present paper to sketch another simple proof of Theorem 1 using Poincar´e’s m´ethode de balayage. For this purpose we consider a sequence {uk } of functions constructed by this method from g. Using §§12–18 in Kellogg [4] we obtain uniform convergence of {uk } on any compactum of Ω. The limit u of the uk is harmonic in Ω and can be extended to Ω ∪ Σ as a continuous function such that u(x0 ) = g(x0 ) for x0 ∈ Σ. On the other hand, we obtain D(uk+1 ) ≤ D(uk ) ≤ D(g) for any k ∈ N provided that the sweeping-out method uses smoothly bounded subdomains for the construction of the uk . It follows that D(u) < ∞ and u ∈ C(Ω, g), and u is seen to be the minimizer of D in C(Ω, g). This completes the proof of Theorem 1. The reader might welcome a somewhat more detailed presentation of this proof avoiding the reference to [4]; this will be carried out in the sequel. Let Ω be a nonempty bounded domain in Rn , and denote by S(Ω) the class of subharmonic functions v ∈ 0 C (Ω) which are defined as solutions of the mean value inequality v(x0 ) ≤

 \

v(x) dx

BR (x0 ) ∗

e-mail: [email protected], Phone: +49 (0)2241 330988 c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Hildebrandt: On Dirichlet’s principle and Poincar´e’s m´ethode de balayage

that is to hold for all balls BR (x0 ) ⊂⊂ Ω. Similarly T(Ω) denotes the class of superharmonic functions w ∈ C 0 (Ω) satisfying  w(x0 ) ≥ \ w(x) dx . BR (x0 )

According to Koebe’s
theorem, H(Ω) := S(Ω) ∩ T(Ω) is the class of functions which are harmonic in Ω.  A function b ∈ C 0 Ω ∩ T(Ω) is called a barrier for Ω (and ∆) at x0 ∈ ∂Ω if b(x0 ) = 0 and b(x) > 0 for x ∈ Ω\{x0 }; in this case x0 is called a regular boundary point of Ω. By Σ we denote the set of regular boundary points of Ω. 0 0 n Fix an arbitrary
g ∈ C (∂Ω). By Tietze’s theorem we can assume that g ∈ C (R ) and supRn |g| ≤ 0 sup∂Ω |g|. On C Ω we introduce the norm |u|0 := supΩ |u| and the seminorms |u|0,S := supS |u| for S ⊂ Ω.
 Using Poisson’s integral theorem we can associate with any ball B ⊂⊂ Ω a linear mapping PB of C 0 Ω into itself which is defined by (PB v)(x) := v(x) for any x ∈ Ω \B and (PB v)(x) := h(x) for x ∈ B where h is the uniquely determined element in H(B) with h = v on ∂B. The operator PB is a projector, i.e. PB2 = PB , and we have |PB |0 = 1 on account of the maximum principle. Moreover, PB leaves the spaces S(Ω) and T(Ω) invariant, and we have v ≤ PB v for v ∈ S(Ω) and PB w ≤ w for w ∈ T(Ω). Poincar´e’s sweeping-out method is the following iterative procedure: We choose a sequence of open balls Bi ⊂⊂ Ω, i ∈ N, such that Ω = B1 ∪ B2 ∪ . . . ∪ Bi ∪ . . . , and consider the projectors Pi := PBi . Then we form the
sequence of indices ik ∈ N given by 12123123412345 . . ., and define the sequence {uk } of functions  uk ∈ C 0 Ω with uk = g on ∂Ω by u0 := g ,

uk := Pik uk−1

for k ∈ N .

(1)

Theorem 2 There is some u ∈ H(Ω) such that uk (x) ⇒ u(x)

in any Ω ⊂⊂ Ω

(2)

i.e. the convergence is uniform on every compactum in Ω. Moreover, u can be extended continuously to Ω ∪ Σ with u(x0 ) = g(x0 )

for x0 ∈ Σ .

(3)

P r o o f. (i) Suppose also that g is subharmonic. Then uk ∈ S(Ω) for any k ∈ N, and g ≤ u1 ≤ u2 ≤ u3 ≤ . . . ≤ m := sup g . ∂Ω

Hence u(x) := limk→∞ uk (x) exists for every x ∈ Ω. Fix some index j
∈ N. Since j appears infinitely often in uk |Bj ∈ H Bj . By Harnack’s theorem it follows the sequence {ik }, there is a subsequence {uk } of {uk } with
that uk (x) ⇒ u(x) in every B  ⊂⊂ Bj whence u|Bj ∈ H Bj and uk (x) ⇒ u(x) in B  ⊂⊂ Bj . Hence we obtain (2) and u ∈ H(B).
 (ii) Now we drop the assumption g ∈ S(Ω), but assume g ∈ C 2 Ω . Then there is a number c > 0 with
 c ∆g > −c. Set g1 (x) := 2n |x|2 for x ∈ Ω and g2 := g + g1 ; clearly, g1 , g2 ∈ C 0 Ω ∩ S(Ω). If we form v0 := g1 , w0 := g2 , and vk := Pik vk−1 , wk := Pik wk−1 for k ≥ 1, we obtain by (i) that vk (x) ⇒ v(x) and wk (x) ⇒ w(x) in Ω ⊂⊂ Ω as well as v, w ∈ H(Ω). Since g = g2 − g1 we have uk = wk − vk for the sequence {uk } defined by (2); thus u := v − w ∈ H(Ω) and uk (x) ⇒ u(x) in Ω ⊂⊂ Ω. (iii) Finally we consider an arbitrary g ∈ C 0 (∂Ω), extended to g ∈ C 0 (Rn ). Mollifying g, we obtain   functions σ g ∈ C ∞ (Rn ), σ > 0, with g σ (x) ⇒ g(x) on Ω as σ → 0. For any σ > 0 we form the sequence uσk as in (1): uσ0 := g σ , uσk := Pik uσk−1 for k ≥ 0. By (ii) we have |uσk − uσl |0,Ω
−→ 0 as k , l −→ ∞ for any Ω ⊂⊂ Ω .

(4)

Set Ak := Pik Pik−1 . . . Pi1 ; then |Ak |0 ≤ |Pik |0 . . . |Pi1 |0 ≤ 1. From uk = Ak g and uσk = Ak g σ it follows that uk − uσk = Ak (g − g σ ) whence   uk − uσk  ≤ |Ak |0 |g − g σ |0 ≤ |g − g σ |0 . 0 c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


Math. Nachr. 278, No. 1–2 (2005) / www.mn-journal.com

143

Thus, for Ω ⊂⊂ Ω,

      |uk − ul |0,Ω
≤ uσk − uσl 0,Ω
+ uk − uσk 0 + ul − uσl 0   ≤ uσk − uσl 0,Ω
+ 2 |g − g σ |0 .

From (4) and |g − g σ |0 → 0 as σ → 0 we infer: Given ε > 0 and Ω ⊂⊂ Ω, there is an N ∈ N such that |uk − ul |0,Ω
< ε

for k , l > N .

Then, by Weierstrass’s theorem, there is a u ∈ H(Ω) such that (2) is satisfied. (iv) Let x0 ∈ Σ. Given ε > 0 there is a constant c > 0 such that |g(x) − g(x0 )| ≤ ε + cb(x)

for all x ∈ Ω .

Then the functions v := g(x0 )−ε−cb and w := g(x0 )+ε+cb are of class S(Ω) and T(Ω) respectively and satisfy v ≤ g ≤ w. The maximum principle implies v ≤ u1 ≤ w, and in the same way v ≤ u2 ≤ w, v ≤ u3 ≤ w, etc. . In general, v ≤ uk ≤ w for any k ∈ N, and so (2) leads to v ≤ u ≤ w on Ω, that is, g(x0 ) − ε − cb(x) ≤ u(x) ≤ g(x0 ) + ε + cb(x)

for any x ∈ Ω .

Since b(x) → 0 as x → x0 , we obtain g(x0 ) − ε ≤ lim inf u(x) ≤ lim sup u(x) ≤ g(x0 ) + ε x→x0

x→x0

for any ε > 0 whence u(x) → g(x0 ) as x → x0 . In order to verify Theorem 1, the following three well-known results are used: Proposition 3 For any g ∈ H 1 (Ω) there exists exactly one minimizer v of D in C(Ω, g). This minimizer is harmonic in Ω, and every minimizing sequence of D in C(Ω, g) converges strongly in H 1,2 (Ω) to v. Proposition 4 If u ∈ H 1 (Ω) ∩ C 2 (Ω) is harmonic in Ω then D(u) < D(v) for all v ∈ H 1 (Ω) with v = u ◦

and v − u ∈ H 1 (Ω).


 Proposition 5 If Ω is a smoothly bounded domain, e.g. a ball, and g ∈ H 1 (Ω) ∩ C 0 Ω then the minimizer
 u of D in C(Ω, g) is of class H 1 (Ω) ∩ C 0 Ω . P r o o f of Theorem 1. From Proposition 4 we infer that the sequence {uk } defined by (1) satisfies D(g) ≥ D(u1 ) ≥ D(u2 ) ≥ . . . . Since |uk |0 ≤ |g|0 for all k ∈ N, the sequence {uk } is bounded in H 1 (Ω), and so we can extract a subsequence that converges weakly in H 1 (Ω) to some v ∈ H 1 (Ω). By (2) it follows that v = u, and so u lies in H 1 (Ω).

{uk }

◦

◦

From uk − g ∈ H 1 (Ω) we then infer u − g ∈ H 1 (Ω), i.e. u ∈ C(Ω, g) ∩ H(Ω). Now Proposition 4 implies that u is the minimizer of D in C(Ω, g). Taking Proposition 3 and Theorem 2 into account we arrive at Theorem 1.  Remark It is not a priori clear that {uk } is a minimizing sequence for D in C(Ω, g). However, if one replaces the sequence {ik } of indices in (1) chosen as 12 123 1234 12345 . . . by the more symmetric sequence 1213121412131215121312141213121 . . . it is possible to prove uk − u1 → 0 whence D(uk ) → D(u), and so {uk } is even a minimizing sequence in C(Ω, g) (see F. Stummel [7]). c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Hildebrandt: On Dirichlet’s principle and Poincar´e’s m´ethode de balayage

References [1] J. Frehse, Capacity Methods in the Theory of Partial Differential Equations, Jber. d. Dt. Math.-Ver. 84, 1–44 (1982). [2] O. Frostman, Les points irreguliers dans la th´eorie du potentiels et le crit`ere de Wiener, Fysiografiska S¨allskapets J. Lund F¨orhandlingar 9 (1939). [3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der math. Wiss. (Springer-Verlag, Berlin, 1977). [4] O. D. Kellogg, Foundations of Potential Theory. Reprint: Dover Publications (New York, 1953), First Edition: SpringerVerlag (Berlin, 1929). [5] W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Normale Sup. Pisa 17, Ser. III, 45–79 (1963). [6] C. G. Simader, Equivalence of weak Dirichlet’s principle, the method of weak solutions and Perron’s method towards classical solutions of Dirichlet’s problem for harmonic functions, to appear in Math. Nachr. [7] F. Stummel, Zur Konvergenz des Balayage-Verfahrens in Hilbertschen R¨aumen, Math. Z. 86, 136–144 (1964).

c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim