Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

Lemma A.1.

Let u be a regular solution of - Δ p ⁢ u = f ⁢ ( u ) in B R ⊂ R n , with f ∈ C 1 ⁢ ( R ) nonnegative. Then, for every ρ ∈ ( 0 , R ) , we have

where C > 0 is a constant depending only on n.

Proof.

First, we recall that since ∇ ⁡ u ∈ W σ , loc 1 , 2 ⁢ ( B R ) – see Remark 2.2 – we can take the distributional divergence of | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u .

We choose a nonnegative function η ∈ C c 1 ⁢ ( B R ) such that η ≡ 1 in B ρ and | ∇ ⁡ η | ≤ C R - ρ . Then, since - Δ p ⁢ u ≥ 0 , we have

and this concludes the proof. ∎

Lemma A.2.

Let p ≥ 2 and let ( w k ) k be a sequence of weakly differentiable functions in L p - 1 ⁢ ( B 1 ) such that

for some constant C independent of k. Then there exists a subsequence of ( w k ) k that converges strongly in L p - 1 ⁢ ( B 1 ) .

Proof.

For w ~ k := | w k | p - 2 ⁢ w k we have that

where C does not depend on k. Thus, the sequence ( w ~ k ) k is bounded in W 1 , 1 ⁢ ( B 1 ) and from the compactness of the immersion W 1 , 1 ⁢ ( B 1 ) ↪ L 1 ⁢ ( B 1 ) we obtain that, up to a subsequence,

for some w ~ ∈ L 1 ⁢ ( B 1 ) .

Now, defining w := | w ~ | p ′ - 2 ⁢ w ~ , we have that

where we used that for p ≥ 2 we have | a - b | p ≤ C n , p ⁢ ( | a | p - 2 ⁢ a - | b | p - 2 ⁢ b ) ⋅ ( a - b ) if a and b belong to ℝ n ; see [19, Chapter 1, Lemma 4.4]. As a consequence, up to subsequences, from the strong convergence of w ~ k to w ~ in L 1 ⁢ ( B 1 ) we deduce that

This finishes the proof. ∎

Proposition A.3.

Let p ≥ 2 , ε ∈ ( 0 , 1 ) , let B 1 be the unit ball of R n , and let u ∈ ( C 1 ∩ W 1 , p - 1 ) ⁢ ( B 1 ) satisfy ∇ ⁡ u ∈ W σ 1 , 1 ⁢ ( B 1 ) with σ = | ∇ ⁡ u | p - 2 . Then

where C depends only on n and p.

Proof.

First, we observe that the regularity assumptions on u ensure that all the integrals in (A.1) are well defined.

We first prove (A.1) in dimension n = 1 . Given ε > 0 , let u have the regularity assumed in the statement with B 1 replaced by ( 0 , ε ) . We claim that

where the constant C > 0 depends only on p.

The inequality is trivial if inf ( 0 , ε ) ⁡ | u ′ | = 0 . Therefore, we assume that inf ( 0 , ε ) ⁡ | u ′ | > 0 and, by Bolzano’s theorem, either u ′ > 0 in ( 0 , ε ) or u ′ < 0 in ( 0 , ε ) . By eventually changing u to - u , we may assume that u ′ > 0 in ( 0 , ε ) . For a , b ∈ ℝ satisfying 0 < a < ε 4 < 3 ⁢ ε 4 < b < ε , we have

Integrating this inequality in b ∈ ( 3 ⁢ ε 4 , ε ) , we get

Integrating now in a ∈ ( 0 , ε 4 ) ,

Thus

and raising this inequality to p - 1 ≥ 1 , we get

where C depends only on p. This proves (A.2).

Therefore, by (A.2), there exists a point x 0 ∈ ( 0 , ε ) such that

Let x ∈ ( 0 , ε ) and let I 0 ⊂ ( 0 , ε ) be the interval with end points x 0 and x. Then

and hence

Combining this inequality with (A.3) and integrating in x ∈ ( 0 , ε ) , we obtain

Note that we have not required any specific boundary values for u. Hence, given any bounded open interval I ⊂ ℝ , for k ∈ ℕ we divide I into k disjoint intervals of length ε := | I | k . Now, using (A.4) in each of these intervals of length ε and adding up all the inequalities given by (A.4), we deduce

for ε = | I | k . Since k is an arbitrary integer, as large as wished, we conclude that (A.5) holds for every I and ε > 0 . This establishes the proposition in dimension one – after replacing ε by ε p - 1 .

Finally, let B 1 be the unit ball of ℝ n , B 1 n - 1 the unit ball of ℝ n - 1 , u ∈ ( C 1 ∩ W 1 , p - 1 ) ⁢ ( B 1 ) satisfy ∇ ⁡ u ∈ W σ 1 , 1 ⁢ ( B 1 ) , and denote x = ( x 1 , x ′ ) ∈ ℝ × ℝ n - 1 and I x ′ := { x 1 ∈ ℝ : ( x 1 , x ′ ) ∈ B 1 } . Then, using (A.5), we have

where C depends only on p. Since the same inequality holds for the partial derivatives with respect to each variable x k instead of x 1 (and we can replace ε by ε / ( p - 1 ) ), we conclude (A.1). ∎

Lemma A.4.

Let β ∈ R and C 0 > 0 . Let S : B → [ 0 , + ∞ ] be a nonnegative function defined on the class B of open balls B ⊂ R n and satisfying the following subadditivity property:

Assume also that S ⁢ ( B 1 ) < ∞ . Then there exists δ, depending only on n and β, such that if

then

where C depends only on n and β.

Theorem B.1 (Michael-Simon [31] and Allard [2] inequality).

Let M be an ( n - 1 ) -dimensional C 2 -hypersurface of R n , q ∈ [ 1 , n - 1 ) , and let φ ∈ C 1 ⁢ ( M ) have compact support in M. If M is compact without boundary, any function φ ∈ C 1 ⁢ ( M ) is allowed. Then there exists a positive constant C, depending only on n and q, such that

where q * = ( n - 1 ) ⁢ q n - 1 - q is the Sobolev exponent, ∇ T denotes the tangential gradient to M, and H is the mean curvature of M, i.e., the sum of its n - 1 principal curvatures.

Alternative proof of Lemma 3.2.

We divide the proof into three steps.

Step 1. If n ≤ 3 , we can add additional artificial variables and reduce to the case n > 3 , by Remark 1.7. We thus assume that n > 3 and claim that

where C depends only on n and p.

In order to prove (B.2), observe that the surface integral on the left-hand side of (B.2) can be equivalently computed over { u = t } ∩ B 1 / 2 or over { u = t } ∩ B 1 / 2 ∩ { | ∇ u | > 0 } . We use the Michael-Simon and Allard inequality (B.1) on the C 2 hypersurface^[16] M = { u = t } ∩ { | ∇ u | > 0 } . We would like to apply the inequality to the function

where η ∈ C c ∞ ⁢ ( B 3 / 4 ) satisfies 0 ≤ η ≤ 1 and η ≡ 1 in B 1 / 2 ; however, this function will not have, in general, compact support in M = { u = t } ∩ { | ∇ u | > 0 } . Thus, in (B.1) we take the test function

with ε > 0 , where ϕ ∈ C ∞ ⁢ ( ℝ ) takes values into [ 0 , 1 ] , ϕ ⁢ ( t ) = 1 if t ≥ 2 , and ϕ ⁢ ( t ) = 0 if t ≤ 1 . Note that ϕ ⁢ ( | ∇ ⁡ u | ε ) is the cut-off function that we already used in the proof of Lemma 2.3. Using Fatou’s lemma, (B.1) with q = 2 < n - 1 , and the coarea formula, we obtain

(B.3)

Here the tangential gradient ∇ T and the mean curvature H are referred to the level set of u passing through a given point x. For the square modulus of the tangential gradient of φ ε , the Cauchy–Schwarz inequality gives

Plugging this inequality into (B.3), using | ∇ T ⁡ ϕ ⁢ ( | ∇ ⁡ u | ε ) | 2 ≤ | ∇ ⁡ ϕ ⁢ ( | ∇ ⁡ u | ε ) | 2 , (2.7), the fact that | ϕ ′ ⁢ ( | ∇ ⁡ u | ε ) | is bounded independently of ε and it is supported in { ε ≤ | ∇ ⁡ u | ≤ 2 ⁢ ε } , and that | ∇ ⁡ u | 2 ≤ 4 ⁢ ε 2 in this set, we obtain

Now, using (2.4) and dominated convergence, we deduce that the last term is zero. Using that ϕ 2 ≤ 1 in the first integral on the right-hand side of the inequality, we get

Since H 2 ≤ ( n - 1 ) ⁢ | A | 2 , we deduce that

Finally, we use the stability condition in its geometric form, Theorem 1.8, to obtain

This proves (B.2).

Step 2. Now we show that, for almost every t ∈ ℝ ,

where the constants C depend only on n and p. Note that this estimate is similar to (3.24), where we assumed ∥ ∇ ⁡ u ∥ L p ⁢ ( B 1 ) = 1 , but we cannot deduce (B.4) from (3.24) through Hölder’s inequality, since we do not have a control on the measure of { u = t } ∩ B 1 / 2 . However, the proofs of both estimates rely on the same idea.

First, we take η as defined in Step 1 and we claim that, for almost every t ∈ ℝ ,

Observe that u is not regular enough to apply Sard’s theorem and deduce regularity of { u = t } for a.e. t, and hence we cannot integrate by parts in the set { u > t } ∩ B 1 . However, as in the proof of (3.25), we can use a smooth approximation K ε ⁢ ( s ) of the characteristic function of ℝ + , and then send ε ↓ 0 , to establish (B.5).

From (B.5) we obtain that

Now, since 0 ≤ η ≤ 1 has compact support in B 3 / 4 , we can use Lemma A.1 to obtain

Combining this with the previous bound, we conclude the proof of (B.4).

Step 3. Finally, from (B.2) and (B.4) we deduce (3.17). For this, we use Hölder’s inequality with exponents q = n - 1 n - 3 and q ′ = n - 1 2 , as well as the coarea formula, to obtain

This concludes the alternative proof of Lemma 3.2. ∎