Regularity of solutions in semilinear elliptic theory

We study the semilinear Poisson equation \begin{equation} \label{pro} \Delta u = f(x, u) \hskip .2 in \text{in} \hskip .2 in B_1. \end{equation} Our main results provide conditions on $f$ which ensure that weak solutions of this equation belong to $C^{1,1}(B_{1/2})$. In some configurations, the conditions are sharp.


Introduction
The semilinear Poisson equation (1) encodes stationary states of the nonlinear heat, wave, and Schrödinger equation. In the case when f is the Heaviside function in the u-variable, (1) reduces to the classical obstacle problem. For an introduction to classical semilinear theory, see [BS11,Caz06].
It is well-known that weak solutions of (1) belong to the usual Sobolev space W 2,p (B 1/2 ) for any 1 ≤ p < ∞ provided f ∈ L ∞ . Recent research activity has thus focused on identifying conditions on f which ensure W 2,∞ (B 1/2 ) regularity of u.

The classical theory
There are simple examples which illustrate that continuity of f = f (x) does not necessarily imply that u has bounded second derivatives: for p ∈ (0, 1) and x ∈ R 2 such that |x| < 1, the function u(x) = x 1 x 2 (− log |x|) p has a continuous Laplacian but is not in C 1,1 [Sha15]. However, if f is Hölder continuous, then it is well-known that u ∈ C 2,α ; if f is Dini continuous, then u ∈ C 2 [GT01,Kov99]. The sharp condition which guarantees bounded second derivatives of u is the C 1,1 regularity of f * N where N is the Newtonian potential and * denotes convolution; this requirement is strictly weaker than Dini continuity of f .
In the general case, the state-of-the-art is a theorem of Shahgholian [Sha03] which states that u ∈ C 1,1 whenever f = f (x, u) is Lipschitz in x, uniformly in u, and ∂ u f ≥ −C weakly for some C ∈ R. In some configurations this illustrates regularity for continuous functions f = f (u) which are strictly below the classical Dini-threshold in the u-variable, e.g. the odd reflection of about the origin. Shahgholian's theorem is proved via the celebrated Alt-Caffarelli-Friedman (ACF) monotonicity formula and it seems difficult to weaken the assumptions by this method. On the other hand, Koch and Nadirashvili [KN] recently constructed an example which illustrates that the continuity of f is not sufficient to deduce that weak solutions of ∆u = f (u) are in C 1,1 .
We say f = f (x, u) satisfies assumption A provided that f is Dini continuous in u, uniformly in x, and has a C 1,1 Newtonian potential in x, uniformly in u (see §3). One of our main results is the following statement.
Our assumption includes functions which fail to satisfy both conditions in Shahgholian's theorem, e.g.
The proof of Theorem 1.1 does not invoke monotonicity formulas and is self-contained. We consider the L 2 projection of D 2 u on the space of Hessians generated by second order homogeneous harmonic polynomials on balls with radius r > 0 and show that the projections stay uniformly bounded as r → 0 + . Although this approach has proven effective in dealing with a variety of free boundary problems [ALS13, FS14, IM15, IM], Theorem 1.1 illustrates that it is also useful in extending and refining the classical elliptic theory.

Singular case: the free boundary theory
In §4 we study the PDE (1) for functions f = f (x, u) which are discontinuous in the u-variable at the origin.
If the discontinuity of f is a jump discontinuity, (1) has the structure where g 1 , g 2 are continuous functions such that and χ Ω defines the indicator function of the set Ω.
Our aim is to find the most general class of coefficients g i which generate interior C 1,1 regularity.
The classical obstacle problem is obtained by letting g 1 = 1, g 2 = 0, and it is well-known that solutions have second derivatives in L ∞ [PSU12]. Nevertheless, by selecting g 1 = −1, g 2 = 0, one obtains the so-called unstable obstacle problem. Elliptic theory and the Sobolev embedding theorem imply that any weak solution belongs to C 1,α for any 0 < α < 1. It turns out that this is the best one can hope for: there exists a solution which fails to be in C 1,1 [AW06]. Hence, if there is a jump at the origin, C 1,1 regularity can hold only if the jump is positive and this gives rise to: The free boundary Γ = ∂{u = 0} consists of two parts: Γ 0 = Γ ∩ {∇u = 0} and Γ 1 = Γ ∩ {∇u = 0}. The main difficulty in proving C 1,1 regularity is the analysis of points where the gradient of the function vanishes. In this direction we establish the following result.
Theorem 1.2. Suppose g 1 , g 2 satisfy A and B. Then if u is a solution of (1), At points where the gradient does not vanish, the implicit function theorem yields that the free boundary is locally a C 1,α graph for any 0 < α < 1. The solution u changes sign across the free boundary, hence it locally solves the equation ∆u = g 1 (x, u) on the side where it is positive and ∆u = g 2 (x, u) on the side where it is negative. If the coefficients g i are regular enough to provide C 1,1 solutions up to the boundary -this is encoded in assumption C, see §4then we obtain full C 1,1 regularity. Theorem 1.3. Suppose g 1 , g 2 satisfy A, B and C. Let u be a solution of (1) and 0 ∈ Γ 0 . Then u ∈ C 1,1 (B ρ0 (0)), for some ρ 0 > 0.
Equation (1) with right-hand side of the form (2) is a generalization of the well-studied two-phase membrane problem, where g i (x, u) = λ i (x), i = 1, 2. The C 1,1 regularity in the case when λ 1 ≥ 0, λ 2 ≤ 0 are two constants satisfying B was obtained by Uraltseva [Ura01] via the ACF monotonicity formula. Moreover, Shahgholian proved this result for Lipschitz coefficients which satisfy B [Sha03, Example 2]. If the coefficients are Hölder continuous, the ACF method does not directly apply and under the stronger assumption that inf λ 1 > 0 and inf −λ 2 > 0, Edquist, Lindgren, Shahgholian [LSE09] obtained the C 1,1 regularity via an analysis of blow-up limits and a classification of global solutions (see also [LSE09, Remark 1.3]). Theorem 1.3 improves and extends this result.
The difficulty in the case when g i depend also on u is that if v := u + L for some linear function L, then v is no longer a solution to the same equation, so one has to get around the lack of linear invariance. Our technique exploits that linear perturbations do not affect certain L 2 projections.
The proof of Theorem 1.3 does not rely on classical monotonicity formulas or classification of global solutions. Rather, our method is based on an identity which provides monotonicity in r of the square of the L 2 norm of the projection of u onto the space of second order homogeneous harmonic polynomials on the sphere of radius r.
Theorems 1.2 & 1.3 deal with the case when f has a jump discontinuity. If f has a removable discontinuity, (1) has the structure ∆u = g(x, u)χ u =0 . ( In this case, one may merge some observations in the proofs of the previous results with the method in [ALS13] and prove the following theorem. Theorem 1.4. If g satisfies assumption A, then every solution of (3) is in C 1,1 (B 1/2 ).
Theorems 1.1 -1.4 provide a comprehensive theory for the general semilinear Poisson equation where the free boundary theory is encoded in the regularity assumption of f in the u-variable.

Technical tools
Throughout the text, the right-hand side of (1) is assumed to be bounded. Moreover, P 2 denotes the space of second order homogeneous harmonic polynomials. A useful elementary fact is that all norms on P 2 are equivalent.
Lemma 2.1. The space P 2 is a finite dimensional linear space. Consequently, all norms on P 2 are equivalent.
For u ∈ W 2,2 (B 1 ), y ∈ B 1 and r ∈ (0, dist(y, ∂B 1 )), Π y (u, r) is defined to be the L 2 projection operator on P 2 given by Calderon-Zygmund theory yields the following useful inequality for re-scalings of weak solutions of (1).
Our analysis requires several additional simple technical lemmas involving the projection operator.
Proof. Let f = ∆u and v be the Newtonian potential of f , i.e. v where ω n is the volume of the unit ball in R n . Since u − v is harmonic, Invoking bounds on the projection (e.g. [ALS13, Lemma 3.2]) and Calderon-Zygmund theory (e.g. [ALS13, Theorem 2.2]), it follows that The L ∞ bound follows from the equivalence of the norms in the space P 2 .
Proof. Note that As for the second inequality in the statement of the lemma let r 0 = 1/4 and s ∈ [1/2, 1]. Then we have that for all j ≥ 1.
The previous tools imply a growth estimate on weak solutions solution of (1).
Lemma 2.5. Let u solve (1). Then for y ∈ B 1/2 and r > 0 small enough, Proof. Letũ The assertion of the Lemma is equivalent to the estimate for r small enough. Lemma 2.4 and the C 1,α estimates of Lemma 2.2 imply provided r is small enough.
Next lemma relates the boundedness of the projection operator and the boundedness of second derivatives of weak solutions of (1).
Lemma 2.6. Let u be a solution to (1). If for each y ∈ B 1/2 there is a sequence Proof. Let y ∈ B 1/2 be a Lebesgue point for D 2 u and r j = r j (y) → 0 + as j → ∞. Then by utilizing Lemma 2.2, Since a.e. z ∈ B 1/2 is a Lebesgue point for D 2 u, the proof is complete.
Next, we introduce another projection that we need for our analysis. Define Q y (u, r) to be the minimizer of The following lemma records the basic properties enjoyed by this projection, cf.
i. This is evident.
ii. It suffices to prove Q y (u, r) = Q y (u, 1) for r < 1. Let and for i = 2, let σ i be an i th degree harmonic polynomial. Then there exist coefficients a i such that Then v is a harmonic and u(x + y) = v(x) for x ∈ ∂B 1 . Hence, we have that u(x + y) = v(x) for x ∈ B 1 and in particular iii. & iv. These are evident.
v. Similar to Lemma 2.3.
vi. This follows from the fact that Q 0 (u, 1) is the L 2 projection of u.
Next we prove some technical results for Q y (u, r) and establish a precise connection between Π y (u, r) and Q y (u, r) by showing that the difference is uniformly bounded in r.
Lemma 2.8. For u ∈ W 2,p (B 1 (y)) with p large enough and r ∈ (0, 1], Proof. Firstly, Since u is C 1,α if p large enough and Q is linear bounded operator, it follows that Lemma 2.9. Let u ∈ W 2,p (B 1 (y)) with p large enough and q ∈ P 2 . Then Proof. Integration by parts implieŝ By taking into account that q is a second order homogeneous polynomial it follows that ∂q(x) ∂n = 2q(x), x ∈ ∂B 1 .
Proof. i. For each r, the difference u r − v r = Q y (u, r) − Π y (u, r) is a second order harmonic polynomial. Therefore, it suffices to show that L ∞ norm of that difference admits a bound independent of r. Note that Hence, ii. Lemma 2.2 implies that {u r } r>0 is bounded in C 1,α (B 1 )∩W 2,p (B 1 ) for every α < 1 and p > 1. Hence, the result follows from i.

C 1,1 regularity: general case
In this section we utilize the previous technical tools and prove C 1,1 regularity provided that f = f (x, t) satisfies assumption A: Assumption A.
where h ∈ L ∞ (B 1 ) and ǫ 0 ω(t) t dt < ∞, for some ǫ > 0; (ii) The Newtonian potential of x → f (x, t) is C 1,1 locally uniformly in t: for v t := f (·, t) * N where N is the Newtonian potential, Proof of Theorem 1.1.
To generate examples, consider f (x, t) = φ(x)ψ(t). If φ ∈ L ∞ and ψ is Dini, then f satisfies condition (i). If φ * N is C 1,1 and ψ is locally bounded, then f satisfies (ii). Thus if φ * N is C 1,1 and ψ is Dini, then f satisfies both conditions. In particular, f may be strictly weaker than Dini in the x-variable.
Remark 2. The projection Q y has similar properties to Π y . Consequently, if f satisfies assumption A, (5) holds for Π y replaced by Q y .

C 1,1 regularity: discontinuous case
The goal of this section is to investigate the optimal regularity for solutions of (1) with f having a jump discontinuity in the t-variable. This case may be viewed as a free boundary problem. The idea is to employ again an L 2 projection operator.

Two-phase obstacle problem
where g 1 , g 2 are continuous. We recall from the introduction that if f has a jump in u at the origin, then we assume it to be a positive jump: Remark 3. In the unstable obstacle problem, i.e. g 1 = −1, g 2 = 0, there exists a solution which is C 1,α for any α ∈ (0, 1) but not C 1,1 .
The proof of the theorem is carried out in several steps. A crucial ingredient is the following monotonicity result.
Lemma 4.2. Suppose g 1 , g 2 ∈ C 0 satisfy B. Then for all constants θ, M > 0 there exist κ 0 (θ, M, g 1 ∞ , g 2 ∞ , n) > 0 and r 0 (θ, M, g 1 ∞ , g 2 ∞ , n) > 0 such that for any solution u of (1) with u L ∞ (B1) ≤ M if Q y (u, r) L 2 (∂B1) ≥ κ 0 , for some 0 < r < r 0 and y ∈ B 1/2 ∩ Γ ∩ {|∇u(y)| < θr}, then Proof. If the conclusion is not true, then there exist radii r k → 0, solutions u k and points and consider the sequence Without loss of generality we can assume that y k → y 0 for some y 0 ∈ B 1/2 . Lemma 2.2 implies the existence of a function v such that up to a subsequence Evidently, v(y 0 ) = |∇v(y 0 )| = 0. Moreover, for q k (x) := Q y k (u k , r k )/T k , we can assume that up to a further subsequence, q k → q in C ∞ for some q ∈ P 2 . Note that By Lemma 2.11, On the other hand Proof of Theorem 4.1. Let κ 0 and r 0 be the constants from Lemma 4.2.
Theorem 4.1 implies C 1,1 regularity away from Γ 1 in the case the coefficients g i are regular enough to provide C 1,1 solutions away from the free boundary, i.e. Theorem 1.2.
Remark 4. Note that A is the condition given in Theorem 1.1. If g i only depend on x, then this reduces to the assumption that the Newtonian potential of g i is C 1,1 , which is sharp.
In this case B δ/2 ∩ Γ = ∅, hence u satisfies the equation u) in B δ/2 (y) for i = 1 or i = 2. Inequality (5) in the Theorem 1.1 assumption A yields Let w ∈ Γ 0 be such that d := |y − w| = dist(y, Γ 0 ). We have that d ≤ δ/2. As before, assumption A yields for r ≤ d/2. From Theorem 4.1 we have that for all |z| ≤ 1 because d ≤ δ/2 ≤ r 0 . On the other hand where Proj P2 is the L 2 (∂B 1 (0)) projection on the space P 2 . We have used the fact that the projection of a linear function is 0. Hence The proof is now complete.
Lastly we point out that if the coefficients g i are regular enough to provide C 1,1 solutions at points where the gradient does not vanish, then we obtain full interior C 1,1 regularity.
Assumption C. For any M > 0 there exist θ 0 (M, g 1 ∞ , g 2 ∞ , n) > 0 and C 3 (M, g 1 ∞ , g 2 ∞ , n) > 0 such that for all z ∈ B 1/2 any solution of Remark 5. A sufficient condition which ensures C is that g i are Hölder continuous, see [LSE09, Proposition 2.6] and [ADN64, Theorem 9.3]. The idea being that at such points, the set {u = 0} is locally C 1,α (via the implicit function theorem) and one may thereby reduce the problem to a classical PDE for which up to the boundary estimates are known.
Theorem 4.1 and C imply Theorem 1.3.
From Corollary 1.2 we can assume that 2d < r 0 . One of the following cases is possible.
In this case we have that (11) holds for r ≤ r 0 by Theorem 4.1.
Here, (11) follows from the assumption C.
By assumption A we have that Hence from (12) we get The previous analysis applies to the following example.
We recall from the introduction that under the stronger assumption inf B1 λ 1 > 0, inf B1 −λ 2 > 0, this problem is studied in [LSE09] and the optimal interior C 1,1 regularity is established. The authors use a different approach based on monotonicity formulas and an analysis of global solutions via a blow-up procedure.

No-sign obstacle problem
Here we observe that assumption A implies that the solutions of (3) are in C 1,1 (B 1/2 ). This theorem was proven in [ALS13] (Theorem 1.2) for the case when g(x, t) depends only on x. Under assumption A, appropriate modifications of the proof in [ALS13] work also for the general case; since the arguments are similar, we provide only a sketch of the proof and highlight the differences. The proof of Theorem 1.2 in [ALS13] consists of the following ingredients.
• Interior C 1,1 estimate • Quadratic growth away from the free boundary Let us recall that the interior C 1,1 estimate is the inequality where ∆u(x) = g(x) for x ∈ B d and the Newtonian potential of g is C 1,1 . This estimate is purely a consequence of g having a C 1,1 Newtonian potential. Quadratic growth away from the free boundary is a bound The first observation in [ALS13] is that if g(x, t) = g(x) has a C 1,1 Newtonian potential, then (14) and (13) yield C 1,1 regularity for the solution. Indeed, "far" from the free boundary, the solution u solves the equation ∆u = g(x) and is locally C 1,1 by assumption. For points close to the free boundary, u solves the same equation but now on a small ball centered at the point of interest and touching the free boundary. At this point one invokes (14) and by (13) obtains that the C 1,1 bound does not blow up close to the free boundary (see Lemma 4.1 in [ALS13]). To prove (14), the authors prove in Proposition 5.1 [ALS13] that if the projection Π y (u, r) (for some y ∈Γ) is large enough then the density λ r of the coincidence set diminishes at an exponential rate. On the other hand, if λ r diminishes in an exponential rate, Π y (u, r) has to be bounded. Consequently, by invoking Lemma 2.2 one obtains (14). Now let g satisfy A.
• Interior C 1,1 estimate In the general case, (13) is replaced by where 0 < s < r < d, ∆v u(y) = g(x, u(y)) and ∆u = f (x, u) in B d (y). Estimate (15) is purely a consequence of assumption A (see (5) in the proof of Theorem 1.1).
• [ALS13, Proposition 5.1] In this proposition, it is shown that there exists C such that if Π y (u, r) ≥ C then for someC > 0. The inequality is obtained by the decomposition u(rx + y) r 2 = Π y (u, r) + h r + w r , where h r , w r are such that ∆h r = −g(rx + y)χ Λr in B 1 , h r = 0 on ∂B 1 , and ∆w r = g(rx + y) in B 1 , w r = u(rx+y) r 2 − Π y (u, r) on ∂B 1 .
• Quadratic growth away from the free boundary In [ALS13], the norms of Π y (u, r/2 k ), k ≥ 1 are estimated in terms of the sum ∞ j=0 λ r/2 j . If the norms of projections are unbounded, one obtain estimate (16) which implies convergence of the previous sum and hence boundedness of the projections. This is a contradiction.
Similarly, in the general case the norms of Q y (u, r/2 k ), k ≥ 1 can be estimated by ∞ j=0 λ r/2 j + ∞ j=0 ω r 2 k 2 log 2 k r 2 .

Inequality (19) and Dini continuity imply
if the norms of projections are unbounded. Furthermore, one completes the proof of the quadratic growth as in [ALS13].
So there exists a constant C such that for all y ∈ B 1/2 there exist radii r j (y) → 0 such that Q y (u, r j (y)) ≤ C.
We conclude via Lemma 2.6.