Abstract
We study the semilinear Poisson equation
Our main results provide conditions on f which ensure that weak solutions of (1) belong to \(C^{1,1}(B_{1/2})\). In some configurations, the conditions are sharp.
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1 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 uvariable, (1) reduces to the classical obstacle problem. For an introduction to classical semilinear theory, see [4, 5].
It is wellknown that weak solutions of (1) belong to the usual Sobolev space \(W^{2,p}(B_{1/2})\) for any \(1\le p<\infty \) provided \(f \in L^\infty \). Recent research activity has thus focused on identifying conditions on f which ensure \(W^{2,\infty }(B_{1/2})\) regularity of u.
1.1 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 \(p \in (0,1)\) and \(x \in \mathbb {R}^2\) such that \(x<1\), the function
has a continuous Laplacian but is not in \(C^{1,1}\) [15]. However, if f is Hölder continuous, then it is wellknown that \(u \in C^{2,\alpha }\); if f is Dini continuous, then \(u \in C^2\) [7, 11]. 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 stateoftheart is a theorem of Shahgholian [14] which states that \(u \in C^{1,1}\) whenever \(f=f(x,u)\) is Lipschitz in x, uniformly in u, and \(\partial _u f \ge C\) weakly for some \(C \in \mathbb {R}\). In some configurations this illustrates regularity for continuous functions \(f=f(u)\) which are strictly below the classical Dinithreshold in the uvariable, 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 [10] recently constructed an example which illustrates that the continuity of f is not sufficient to deduce that weak solutions of \(\Delta u = f(u)\) are in \(C^{1,1}\). With all this in mind, we make the following assumption.
Assumption A
Let \(f=f(x,u)\) be Dini continuous in u, uniformly in x, and assume it has a \(C^{1,1}\) Newtonian potential in x, uniformly in u.
One of our main results is the following statement.
Theorem 1.1
Suppose f satisfies Assumption A. Then any solution of (1) is \(C^{1,1}\) in \(B_{1/2}\).
Our assumption includes functions which fail to satisfy both conditions in Shahgholian’s theorem, e.g.
for \(p>1\), \(x=(x_1,x_2) \in B_1\) and \(t\in (1,1)\). The Newtonian potential assumption in the xvariable is essentially sharp whereas the condition in the tvariable is in general not comparable with Shahgholian’s assumption.
The proof of Theorem 1.1 does not invoke monotonicity formulas and is selfcontained. We consider the \(L^2\) projection of \(D^2u\) 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 \(r \rightarrow 0^+\). Although this approach has proven effective in dealing with a variety of free boundary problems [2, 6, 8, 9], Theorem 1.1 illustrates that it is also useful in extending and refining the classical elliptic theory.
1.2 Singular case: the free boundary theory
In §4 we study the PDE (1) for functions \(f=f(x,u)\) which are discontinuous in the uvariable 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 \(\chi _\Omega \) defines the indicator function of the set \(\Omega \).
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 wellknown that solutions have second derivatives in \(L^\infty \) [13]. Nevertheless, by selecting \(g_1=1,g_2=0\), one obtains the socalled unstable obstacle problem. Elliptic theory and the Sobolev embedding theorem imply that any weak solution belongs to \(C^{1,\alpha }\) for any \(0<\alpha <1\). It turns out that this is the best one can hope for: there exists a solution which fails to be in \(C^{1,1}\) [3]. 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:
Assumption B
\(g_1(x,0)g_2(x,0) \ge \sigma _0,\ x \in B_1\) for some \(\sigma _0>0\).
The free boundary \(\Gamma =\partial \{u \ne 0\}\) consists of two parts: \(\Gamma ^0=\Gamma \cap \{\nabla u = 0\}\) and \(\Gamma ^1=\Gamma \cap \{\nabla u \ne 0\}\). 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), \(\Vert u\Vert _{C^{1,1}(K)}<\infty \) for any \(K \Subset B_{1/2}(0){\setminus } \overline{\Gamma ^1}\).
At points where the gradient does not vanish, the implicit function theorem yields that the free boundary is locally a \(C^{1,\alpha }\) graph for any \(0<\alpha <1\). The solution u changes sign across the free boundary, hence it locally solves the equation \(\Delta u =g_1(x,u)\) on the side where it is positive and \(\Delta u =g_2(x,u)\) 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—then we obtain full \(C^{1,1}\) regularity.
Assumption C
For any \(M>0\) there exist \(\theta _0(M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)>0\) and \(C_3(M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)>0\) such that for all \(z\in B_{1/2}\) any solution of
admits a bound
Remark 1
A sufficient condition which ensures C is that \(g_i\) are Hölder continuous, see [12, Proposition 2.6] and [1, Theorem 9.3]. The idea being that at such points, the set \(\{u=0\}\) is locally \(C^{1,\alpha }\) (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 1.3
Suppose \(g_1,g_2\) satisfy A, B and C. Let u be a solution of (1) and \(0\in \Gamma ^0\). Then \(u \in C^{1,1}(B_{\rho _0}(0))\), for some \(\rho _0>0\).
Equation (1) with righthand side of the form (2) is a generalization of the wellstudied twophase membrane problem, where \(g_i(x,u)=\lambda _i(x),\ i=1,2\). The \(C^{1,1}\) regularity in the case when \(\lambda _1\ge 0\), \(\lambda _2 \le 0\) are two constants satisfying B was obtained by Uraltseva [16] via the ACF monotonicity formula. Moreover, Shahgholian proved this result for Lipschitz coefficients which satisfy B [14, Example 2]. If the coefficients are Hölder continuous, the ACF method does not directly apply and under the stronger assumption that \(\inf \lambda _1>0\) and \(\inf \lambda _2>0\), Edquist, Lindgren, Shahgholian [12] obtained the \(C^{1,1}\) regularity via an analysis of blowup limits and a classification of global solutions (see also [12, 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 and 1.3 deal with the case when f has a jump discontinuity. If f has a removable discontinuity, (1) has the structure
In this case, one may merge some observations in the proofs of the previous results with the method in [2] 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 uvariable.
2 Technical tools
Throughout the text, the righthand side of (1) is assumed to be bounded. Moreover, \(\mathcal {P}_2\) denotes the space of second order homogeneous harmonic polynomials. A useful elementary fact is that all norms on \(\mathcal {P}_2\) are equivalent.
Lemma 2.1
The space \(\mathcal {P}_2\) is a finite dimensional linear space. Consequently, all norms on \(\mathcal {P}_2\) are equivalent.
For \(u \in W^{2,2}(B_1)\), \(y \in B_1\) and \(r \in (0, {{\mathrm{dist}}}(y, \partial B_1))\), \(\Pi _y(u,r)\) is defined to be the \(L^2\) projection operator on \({\mathcal {P}_2}\) given by
Calderon–Zygmund theory yields the following useful inequality for rescalings of weak solutions of (1).
Lemma 2.2
Let u solve (1), \(y \in B_{1/2}\), and \(r\le 1/4\). Then for
it follows that for \(1\le p<\infty \) and \(0<\alpha <1\),
and
Proof
By Calderon–Zygmund theory (e.g. [2, Theorem 2.2]),
in particular,
where \(\overline{D^2 \tilde{u}_r}\) is the average of \(D^2 \tilde{u}_r\) on \(B_{3/2}.\) Now let
and note that this quantity is uniformly controlled by \(\Vert f\Vert _{L^\infty }(B_1\times {\mathbb {R}})\); this fact, and the definition of \(\Pi \) yields (note: \(\text {trace}(\overline{D^2 u}\frac{a}{n}Id)=0\)),
Two applications of Poincaré’s inequality together with the above estimate implies
where the averages are taken over \(B_{3/2}\). Elliptic theory (e.g. [7, Theorem 9.1]) yields that for any \(1\le p<\infty \),
Let \(\phi :=\tilde{u}_r\overline{\nabla \tilde{u}_r} \cdot x \overline{\tilde{u}_r}\). We have that \(\phi (0)=\overline{\tilde{u}_r}\) and \(\nabla \phi (0)=\overline{\nabla \tilde{u}_r}\); however, by the Sobolev embedding theorem, \(\phi \) is \(C^{1,\alpha }\) and thus
completing the proof of the \(W^{2,p}\) estimate. The \(C^{1,\alpha }\) estimate likewise follows from the Sobolev embedding theorem. \(\square \)
Our analysis requires several additional simple technical lemmas involving the projection operator.
Lemma 2.3
For any \(u \in W^{2,2}(B_1)\) and \(s\in [1/2,1]\),
and
for some constant \(C=C(n)\).
Proof
Let \(f=\Delta u\) and v be the Newtonian potential of f, i.e.
where \(\omega _n\) is the volume of the unit ball in \(\mathbb {R}^n\). Since \(uv\) is harmonic,
therefore
Invoking bounds on the projection (e.g. [2, Lemma 3.2]) and Calderon–Zygmund theory (e.g. [2, Theorem 2.2]), it follows that
The \(L^{\infty }\) bound follows from the equivalence of the norms in the space \(\mathcal {P}_2\). \(\square \)
Lemma 2.4
Let u solve (1). Then for all \(0< r \le 1/4,\ s\in [1/2,1]\) and \(y \in B_{1/2}\),
and
for some constant \(C=C(n,\Vert f\Vert _{L^\infty (B_1\times \mathbb {R})},\Vert u\Vert _{L^\infty (B_1)})\).
Proof
Note that
where
as before. From Lemma 2.3 we have that
As for the second inequality in the statement of the lemma let \(r_0=1/4\) and \(s\in [1/2,1]\). Then we have that
for all \(\square \) .
The previous tools imply a growth estimate on weak solutions solution of (1).
Lemma 2.5
Let u solve (1). Then for \(y \in B_{1/2}\) and \(r>0\) small enough,
Proof
Let
The assertion of the Lemma is equivalent to the estimate
for r small enough. Lemma 2.4 and the \(C^{1,\alpha }\) estimates of Lemma 2.2 imply
provided r is small enough. \(\square \)
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 \(y \in B_{1/2}\) there is a sequence \(r_j(y) \rightarrow 0^+\) as \(j \rightarrow \infty \) such that
then
for some constant \(C=C(M,n,\Vert f\Vert _{L^\infty (B_1 \times \mathbb {R})},\Vert u\Vert _{L^{\infty }(B_1)})>0\).
Proof
Let \(y \in B_{1/2}\) be a Lebesgue point for \(D^2 u\) and \(r_j=r_j(y) \rightarrow 0^+\) as \(j \rightarrow \infty \). Then by utilizing Lemma 2.2,
Since a.e. \(z \in B_{1/2}\) is a Lebesgue point for \(D^2 u\), the proof is complete. \(\square \)
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. [2, Lemma 3.2].
Lemma 2.7

(i)
\(Q_y(\cdot ,r)\) is linear;

(ii)
if u is harmonic \(Q_y(u,s)=Q_y(u,r)\) for all \(s<r\);

(iii)
if u is a linear function then \(Q_y(u,r)=0\);

(iv)
if u is a second order homogeneous polynomial then \(Q_y(u,r)=u\);

(v)
\(\Vert Q_0(u,s)Q_0(u,1)\Vert _{L^2(\partial B_1)}\le C_s \Vert \Delta u\Vert _{L^2(B_1)}\), for \(0<s<1\);

(vi)
\(\Vert Q_0(u,1)\Vert _{L^2(\partial B_1)} \le \Vert u\Vert _{L^2(\partial B_1)}\).
Proof

(i)
This is evident.

(ii)
It suffices to prove \(Q_y(u,r)=Q_y(u,1)\) for \(r<1\). Let
$$\begin{aligned} \sigma _2=\frac{Q_y(u,1)}{\Vert Q_y(u,1)\Vert _{L^2(\partial B_1)}} \end{aligned}$$and for \(i \ne 2\), let \(\sigma _i \) be an ith degree harmonic polynomial. Then there exist coefficients \(a_i\) such that
$$\begin{aligned} u(x+y)=\sum \limits _{i=0}^{\infty } a_i \sigma _i(x),\quad x\in \partial B_1; \end{aligned}$$in particular, \(a_2=\Vert Q_y(u,1)\Vert _{L^2(\partial B_1)}\). Let
$$\begin{aligned} v(x)=\sum \limits _{i=0}^{\infty } a_i \sigma _i(x),\quad x\in B_1. \end{aligned}$$Then v is a harmonic and \(u(x+y)=v(x)\) for \(x \in \partial B_1\). Hence, we have that \(u(x+y)=v(x)\) for \(x \in B_1\) and in particular
$$\begin{aligned} u(x+y)=\sum \limits _{i=0}^{\infty } a_i \sigma _i(x),\quad x\in B_1. \end{aligned}$$Therefore
$$\begin{aligned} \frac{u(rx+y)}{r^2}=\sum \limits _{i=0}^{\infty } a_i \frac{\sigma _i(rx)}{r^2}=\sum \limits _{i=0}^{\infty } a_i r^{i2} \sigma _i(x),\quad x\in B_1, \end{aligned}$$so \(Q_y(u,r)=a_2\sigma _2(x)=Q_y(u,1)\).

(iii) and (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.
\(\square \)
We also employ following simple observation in the subsequent analysis.
Remark 2
If H is a Hilbert space and \(E \subset H\) a subspace, then for any \(x \in H\) and \(e \in E\), \(\langle x, e\rangle = \langle Proj_E(x), e\rangle \) (since we may write \(x=Proj_E(x)+y\), where \(y \in E^{\perp }\)).
Next we prove some technical results for \(Q_y(u,r)\) and establish a precise connection between \(\Pi _y(u,r)\) and \(Q_y(u,r)\) by showing that the difference is uniformly bounded in r.
Lemma 2.8
For \(u \in W^{2,p}(B_1(y))\) with \(p>n\) and \(r \in (0,1]\),
Proof
Firstly,
Since u is \(C^{1,\alpha }\) if \(p>n\) and Q is linear bounded operator, it follows that
\(\square \)
Lemma 2.9
Let \(u \in W^{2,p}(B_1(y))\) with \(p>n\) and \(q \in \mathcal {P}_2\). Then
Proof
Integration by parts implies
By taking into account that q is a second order homogeneous polynomial it follows that
Moreover,
Combining these equations yields (4). \(\square \)
Lemma 2.10
Let \(u \in W^{2,p}(B_1(y))\) with \(p>n\) and \(0<r\le 1\). Then for every \(q\in \mathcal {P}_2\),
Proof
Let \(\tilde{u}_r(x)=u(rx+y)/r^2\). From Lemmas 2.8, 2.9, and the fact that \(Q_0(\cdot ,1)\) is the projection onto the space of homogeneous harmonic polynomials of degree two, we obtain
(the third equality follows from Remark 2). \(\square \)
Lemma 2.11
For \(u \in W^{2,p}(B_1(y))\) with \(p>n\) and \(0<r\le 1\),
Proof
\(\square \)
Lemma 2.12
Let \(f\in L^\infty (B_1)\), u be a solution of (1) and \(y \in B_{1/2}\). For \(0<r<1/2\) consider
Then

(i)
\(u_rv_r\) is bounded in \(C^{\infty }\), uniformly in r;

(ii)
the family \(\{v_r\}\) is bounded in \(C^{1,\alpha }(B_1)\cap W^{2,p}(B_1)\), for every \(0<\alpha <1\) and \(p>1\).
Proof
(i) For each r, the difference \(u_rv_r=Q_y(u,r)\Pi _y(u,r)\) is a second order harmonic polynomial. Therefore, it suffices to show that the \(L^{\infty }\) norm of that difference admits a bound independent of r. Note that
Hence, by Lemma 2.2,
(ii) Lemma 2.2 implies that \(\{u_r\}_{r>0}\) is bounded in \(C^{1,\alpha }(B_1)\cap W^{2,p}(B_1)\) for every \(\alpha <1\) and \(p>1\). Hence, the result follows from (i). \(\square \)
3 \(C^{1,1}\) 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

(i)
$$\begin{aligned} f(x,t_2)f(x,t_1)\le \omega (t_2t_1), \end{aligned}$$
and
$$\begin{aligned} \int \limits _{0}^{\epsilon } \frac{\omega (t)}{t}dt<\infty , \end{aligned}$$for some \(\epsilon >0\);

(ii)
The Newtonian potential of \(x \mapsto f(x,t)\) is \(C^{1,1}\) locally uniformly in t: for \(v_t:=f(\cdot ,t) * N\) where N is the Newtonian potential,
$$\begin{aligned} \sup _{a\le t \le b} \Vert D^2 v_t\Vert _{L^\infty (B_1)} < \infty ,\quad \text {for all}\quad a,b \in \mathbb {R}. \end{aligned}$$
Proof of Theorem 1.1
Let \(y \in B_{1/2}\) and \(v=v_{u(y)}=f(x,u(y)) * N\). Note that if
then
Using this identity, Lemmas 2.3 and 2.5
for \(r>0\) sufficiently small (\(\nabla u(y)\) is controlled by \(\Vert u\Vert _{W^{2,p}(B_1)}\)). Hence, for \(r_0>0\) small enough and \(y \in B_{1/2}\) we have
Thus
We conclude via Lemmas 2.4 and 2.6. \(\square \)
Remark 3
To generate examples, consider \(f(x,t)=\phi (x) \psi (t)\). If \(\phi \in L^\infty \) and \(\psi \) is Dini, then f satisfies condition (i). If \(\phi * N\) is \(C^{1,1}\) and \(\psi \) is locally bounded, then f satisfies (ii). Thus if \(\phi * N\) is \(C^{1,1}\) and \(\psi \) is Dini, then f satisfies both conditions. In particular, f may be strictly weaker than Dini in the xvariable.
Remark 4
The projection \(Q_y\) has similar properties to \(\Pi _y\). Consequently, if f satisfies Assumption A, (5) holds for \(\Pi _y\) replaced by \(Q_y\),
4 \(C^{1,1}\) 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 tvariable. This case may be viewed as a free boundary problem. The idea is to employ again an \(L^2\) projection operator.
4.1 Twophase obstacle problem
Suppose \(f=f(x,u)\) has the form
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:
Assumption B
\(g_1(x,0)g_2(x,0) \ge \sigma _0,\ x \in B_1\) for some \(\sigma _0>0\).
Remark 5
In the unstable obstacle problem, i.e. \(g_1=1\), \(g_2=0\), there exists a solution which is \(C^{1,\alpha }\) for any \(\alpha \in (0,1)\) but not \(C^{1,1}\).
Let \(\Gamma ^0:=\Gamma \cap \{\nabla u=u=0\}\) and \(\Gamma ^1:=\Gamma \cap \{\nabla u \ne 0\}\). Our main result provides optimal growth away from points with sufficiently small gradients.
Theorem 4.1
Suppose \(g_1,g_2 \in C^0\) satisfy B. Then for all constants \(\theta ,M>0\) there exist \(r_0(\theta ,M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)>0\) and \(C_0(\theta ,M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)>0\) such that for any solution of (1) with \(\Vert u\Vert _{L^\infty (B_1)} \le M\)
for all y, r such that \(r\le r_0\) and \(y \in B_{1/2} \cap \Gamma \cap \{\nabla u(y) <\theta r\}\). Consequently, for the same choice of r and y we have that
for some constant \(C_1(\theta ,M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)>0\).
The proof of the theorem is carried out in several steps. A crucial ingredient is the following monotonicity result.
Lemma 4.2
Suppose \(g_1,g_2 \in C^0\) satisfy B. Then for all constants \(\theta ,M>0\) there exist \(\kappa _0(\theta , M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)>0\) and \(r_0(\theta ,M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)>0\) such that for any solution u of (1) with \(\Vert u\Vert _{L^\infty (B_1)} \le M\) if
for some \(0<r<r_0\) and \(y \in B_{1/2} \cap \Gamma \cap \{\nabla u(y) <\theta r\}\), then
Proof
If the conclusion is not true, then there exist radii \(r_k \rightarrow 0\), solutions \(u_k\) and points \(y_k \in B_{1/2} \cap \Gamma _k \cap \{\nabla u_k(y_k) <\theta r_k\}\) such that \(\Vert u_k\Vert _{L^\infty (B_1)} \le M\), and \(\Vert Q_{y_k}(u_k,r_k)\Vert _{L^2(\partial B_1)} \rightarrow \infty \), and
Let
and consider the sequence
Without loss of generality we can assume that \(y_k\rightarrow y_0\) for some \(y_0 \in B_{1/2}\). Lemma 2.2 implies the existence of a function v such that up to a subsequence
By \(C^{1,\alpha }\) convergence, \(v(0)=\nabla v(0)=0\). Moreover, for \(q_k(x):=Q_{y_k}(u_k,r_k)/T_k\), we can assume that up to a further subsequence, \(q_k \rightarrow q\) in \(C^{\infty }\) for some \(q \in \mathcal {P}_2\). Note that
hence
By Lemma 2.11,
Therefore
On the other hand
a contradiction. \(\square \)
Proof of Theorem 4.1
Let \(\kappa _0\) and \(r_0\) be the constants from Lemma 4.2. Without loss of generality we can assume that \(r_0\le 1/4\). From Lemmas 2.4 and 2.12 we have that
for all \(y \in B_{1/2}\), where \(C=C(M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)\) is a constant. Let
We claim that
for \(r\le r_0\) and \(y \in B_{1/2} \cap \Gamma \cap \{\nabla u(y) <\theta r\}\). Let us fix y such that \(\nabla u(y)\le \theta r_0\) and consider
as a function of r on the interval \(\nabla u(y)/\theta \le r\le r_0\). Let
By (9) and (10), we have that \(T_y(r_0)\le C_0\), so \(\nabla u(y)/\theta \le e \le r_0\). If \(e>\nabla u(y)/\theta \) then \(T_y(e)=C_0\) and by Lemma 4.2 we have that \(T_{y}'(e)>0\), so \(T_y(r)<C_0\) for \(e\varepsilon<r<e\) which contradicts (11).
Therefore, \(e=\nabla u(y)/\theta \) and \(T_y(r)\le C_0\) for all \(\nabla u(y)/\theta \le r\le r_0\) which proves (7).
Inequality (8) follows from Lemmas 2.2 and 2.12. \(\square \)
Theorem 4.1 implies \(C^{1,1}\) regularity away from \(\Gamma ^1\) 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 6
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.
Proof of Theorem 1.2
Suppose A and B hold. We show that for every \(\delta >0\) there exists \(C_{\delta }>0\) such that for all \(y \in B_{1/2}(0)\) such that \({{\mathrm{dist}}}(y,\Gamma ^1) \ge \delta \), there exists \(r_y>0\) such that
for \(r\le r_y\).
Consequently,
for \(xy\le r_y,\ y\in B_{1/2}(0)\) and \({{\mathrm{dist}}}(y,\Gamma ^1) \ge \delta \); this readily yields the desired result.
Note that (13) follows from (12) via Lemmas 2.2 and 2.12.
Without loss of generality assume that \(\delta \le r_0\), where \(r_0>0\) is the constant from Theorem 4.1. For every \(y \in B_{1/2}(0)\) consider the ball \(B_{\delta /2}(y)\). Then there are two possibilities.

(i)
\(B_{\delta /2}(y) \cap \Gamma ^0=\emptyset \). In this case \(B_{\delta /2} \cap \Gamma =\emptyset \), hence u satisfies the equation
$$\begin{aligned} \Delta u = g_i(x,u) \end{aligned}$$in \(B_{\delta /2}(y)\) for \(i=1\) or \(i=2\). Inequality (6) in Remark 4 and Assumption A yield
$$\begin{aligned} \Vert Q_y(u,r)\Vert _{L^2(\partial B_1(0))}\le C\log \frac{4}{\delta }+C(\Vert D^2v^i_{u(y)}\Vert _{L^\infty (B_1)}+1)\le C_\delta , \end{aligned}$$for \(r\le \delta /4\), where \(v^i_{u(y)}\) is defined as the solution to \(\Delta w(x)=g_i(x,u(y))\).

(ii)
\(B_{\delta /2}(y) \cap \Gamma ^0 \ne \emptyset \). Let \(w\in \Gamma ^0\) be such that \(d:=yw={{\mathrm{dist}}}(y,\Gamma _0)\). We have that \(d\le \delta /2\). As in the previous step, (6) and Assumption A yield
$$\begin{aligned} \Vert Q_y(u,r)\Vert _{L^2(\partial B_1(0))}\le \Vert Q_y(u,d/2)\Vert _{L^2(\partial B_1(0))}+C(\Vert D^2v^i_{u(y)}\Vert _{\infty }+1), \end{aligned}$$for \(r\le d/2\). From Theorem 4.1 we have that
$$\begin{aligned} \left u\left( y+\frac{d}{2}z\right) \right \le C\left y+\frac{d}{2}zw\right ^2\le C d^2, \end{aligned}$$for all \(z\le 1\) because \(d\le \delta /2\le r_0\). On the other hand, by definition,
$$\begin{aligned} Q_y(u,d/2)&={{\mathrm{Proj}}}_{\mathcal {P}_2}\left( \frac{u\left( y+\frac{d}{2}z\right) }{d^2/4}\right) , \end{aligned}$$where \({{\mathrm{Proj}}}_{\mathcal {P}_2}\) is the \(L^2(\partial B_1(0))\) projection on the space \(\mathcal {P}_2\). Hence, by Lemma 2.7 vi,
$$\begin{aligned} \Vert Q_y(u,d/2)\Vert _{L^2(\partial B_1(0))}\le \left\ \frac{u\left( y+\frac{d}{2}z\right) }{d^2/4}\right\ _{L^2(\partial B_1(0))}\le C, \end{aligned}$$which yields
$$\begin{aligned} \Vert Q_y(u,r)\Vert _{L^2(\partial B_1(0))}\le C+C(\Vert D^2v^i_{u(y)}\Vert _{\infty }+1), \end{aligned}$$for \(r\le d/2\).
The proof is now complete. \(\square \)
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. Recall from the introduction the following assumption:
Assumption C
For any \(M>0\) there exist \(\theta _0(M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)>0\) and \(C_3(M,\Vert g_1\Vert _{\infty },\Vert g_2\Vert _{\infty },n)>0\) such that for all \(z\in B_{1/2}\) any solution of
admits a bound
Theorem 4.1 and C imply Theorem 1.3.
Proof of Theorem 1.3
Our strategy is to consider several cases. The main idea is to note that by Lemmas 2.12 and 2.6 the assertion follows if we show that there exist \(\rho _0, C>0\) such that for every \(y \in B_{\rho _0}(0)\) there exists \(r_y>0\) such that
for \(0<r\le r_y\).
Let \(\rho _0\) be such that \(\nabla u(y) \le \theta _0\) for \(y \in B_{\rho _0}(0)\), where \(\theta _0\) is the constant from Assumption C (we can do this because u is \(C^{1,\alpha }\) and \(0 \in \Gamma ^0\)). For \(y \in B_{\rho _0}(0)\) let \(d:={{\mathrm{dist}}}(y,\Gamma )\) and let \(w \in \Gamma \) be such that \(d=yw\).
From Theorem 1.2 we can assume that \(2d<r_0\) where \(r_0\) is the constant in Theorem 4.1. One of the following cases is possible.

(i)
\(d=0, y \in \Gamma ^0\). In this case we have that (14) holds for \(r\le r_0\) by Theorem 4.1 (note that \(\nabla u(y)=0\)).

(ii)
\(d=0, y \in \Gamma ^1\). Here, the \(C^{1,1}\) bound follows directly from Assumption C.

(iii)
\(d>0, w \in \Gamma ^0\). Here, we repeat the argument in case (ii) of the proof of Theorem 1.2 line by line and obtain that (14) is valid for \(r\le d/2\).

(iv)
\(d>0, w \in \Gamma ^1\). From Theorem 4.1 we have that
$$\begin{aligned} u(\tilde{z}+w)\tilde{z}\cdot \nabla u(w) \le C_1 \tilde{z}^2 \end{aligned}$$(15)for \(\nabla u(w)/\theta _0 \le \tilde{z}\le r_0\). On the other hand by Assumption C we obtain that (15) holds for \(\tilde{z} \le \nabla u(w)/\theta _0\). Hence, (15) holds for all \(\tilde{z}\) such that \(\tilde{z}\le r_0\). Thanks to Assumption A, we have via Remark 4 that
$$\begin{aligned} \Vert Q_y(u,r)\Vert _{L^2(\partial B_1(0))}\le \Vert Q_y(u,d/2)\Vert _{L^2(\partial B_1(0))}+C(\Vert D^2v^i_{u(y)}\Vert _{L^{\infty }(B_1)}+1), \end{aligned}$$(16)for \(r\le d/2\). Furthermore, by the projection’s invariance of affine additions
$$\begin{aligned} Q_y(u,d/2)&={{\mathrm{Proj}}}_{\mathcal {P}_2}\left( \frac{u\left( y+\frac{d}{2}z\right) \frac{d}{2}z\cdot \nabla u(y)u(y)}{d^2/4}\right) \\&={{\mathrm{Proj}}}_{\mathcal {P}_2}\left( \frac{u\left( (y+\frac{d}{2}z\right) \left( y+\frac{d}{2}zw\right) \cdot \nabla u(w)}{d^2/4}\right) .\\ \end{aligned}$$Hence by applying (15) with \(\tilde{z}=y+\frac{d}{2}zw\) we have that
$$\begin{aligned} \Vert Q_y(u,d/2)\Vert _{L^2(\partial B_1(0))}&\le \left\ \frac{u\left( w+(y+\frac{d}{2}zw)\right) \left( y+\frac{d}{2}zw\right) \cdot \nabla u(w)}{d^2/4}\right\ _{L^2(\partial B_1(0))}\\&\le C, \end{aligned}$$which combined with (16) yields
$$\begin{aligned} \Vert Q_y(u,r)\Vert _{L^2(\partial B_1(0))}\le C+C(\Vert D^2v^i_{u(y)}\Vert _{L^{\infty }(B_1)}+1), \end{aligned}$$for \(r\le d/2\). Note that \(\tilde{z} \le r_0\) is guaranteed because we have chosen \(d < \frac{r_0}{2}\).
\(\square \)
The previous analysis applies to the following example.
Example
Let \(g_i(x,u)=\lambda _i(x)\) for \(i=1,2\), where \(\lambda _i\) are such that

(i)
\(\lambda _1(x)\lambda _2(x)\ge \sigma _0>0\) for all \(x \in B_1\);

(ii)
\(\lambda _1(x),\lambda _2(x)\) are Hölder continuous.
We recall from the introduction that under the stronger assumption \(\inf _{B_1} \lambda _1>0\), \(\inf _{B_1}\lambda _2>0\), this problem is studied in [12] 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 blowup procedure.
4.2 Nosign 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 [2] (Theorem 1.2) for the case when g(x, t) depends only on x. Under Assumption A, appropriate modifications of the proof in [2] work also for the general case; since the arguments are similar, we provide only a sketch of the proof and highlight the differences.
Sketch of the proof of Theorem 1.4
Let \(\tilde{\Gamma }:=\{y\ \text {s.t.}\ u(y)=\nabla u(y)=0\}\). For \(r>0\) let \(\Lambda _r:=\{x \in B_1\ \text {s.t.}\ u(rx)=0 \}\) and \(\lambda _r:=\Lambda _r\).
The proof of Theorem 1.2 in [2] consists of the following ingredients.

Interior \(C^{1,1}\) estimate

Quadratic growth away from the free boundary

[2, Proposition 5.1]
Let us recall that the interior \(C^{1,1}\) estimate is the inequality
where \(\Delta u (x)= g(x)\) for \(x \in B_d\) and v, 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 [2] is that if \(g(x,t)=g(x)\) has a \(C^{1,1}\) Newtonian potential, then (17) and (18) yield \(C^{1,1}\) regularity for the solution. Indeed, “far” from the free boundary, the solution u solves the equation \(\Delta u=g(x)\) 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 (18) and by (17) obtains that the \(C^{1,1}\) bound does not blow up close to the free boundary (see Lemma 4.1 in [2]).
To prove (18), the authors prove in Proposition 5.1 [2] that if the projection \(\Pi _y(u,r)\) (for some \(y \in \tilde{\Gamma }\)) is large enough then the density \(\lambda _r\) of the coincidence set diminishes at an exponential rate. On the other hand, if \(\lambda _r\) diminishes at an exponential rate, \(\Pi _y(u,r)\) has to be bounded. Consequently, by invoking Lemma 2.2 one obtains (18).
Now let g satisfy A.

Interior \(C^{1,1}\) estimate In the general case, (17) is replaced by
$$\begin{aligned} \Vert Q_y(u,s)\Vert _{L^2(\partial B_1(0))}\le \Vert Q_y(u,r)\Vert _{L^2(\partial B_1(0))}+C(\Vert D^2v_{u(y)}\Vert _{\infty }+1), \end{aligned}$$(19)where \(0<s<r<d,\ \Delta v_{u(y)}=g(x,u(y))\) and \(\Delta u=f(x,u)\) in \(B_d(y)\). Estimate (19) is purely a consequence of Assumption A via Remark 4.

[2, Proposition 5.1]
In this proposition, it is shown that there exists C such that if \(\Pi _y(u,r)\ge C\) then
$$\begin{aligned} \lambda _{r/2}^{1/2}\le \frac{\tilde{C}}{\Vert \Pi _y(u,r)\Vert _{L^\infty (B_1)}}\lambda _{r}^{1/2} \end{aligned}$$(20)
for some \(\tilde{C}>0\). The inequality is obtained by the decomposition
where \(h_r,w_r\) are such that
and
The authors show that
In the general case one may consider the decomposition
where \(h_r,w_r,z_r\) are such that
and
and
Evidently, estimates (21) are still valid. Additionally, we have
since g(x, t) is uniformly Dini in t.
Combining (21) and (22) and arguing as in [2] one obtains the existence of \(C>0\) such that
whenever \(\Vert Q_y(u,r)\Vert _{L^2(\partial B_{1})} \ge C\).

Quadratic growth away from the free boundary In [2], the norms of \(\Pi _y(u,r/2^k),\ k\ge 1\) are estimated in terms of the sum \(\sum \nolimits _{j=0}^{\infty } \lambda _{r/2^j}\). If the norms of projections are unbounded, one obtain estimate (20) 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 \(Q_y(u,r/2^k),\ k\ge 1\) can be estimated by
Inequality (23) and Dini continuity imply
if the norms of projections are unbounded. Furthermore, one completes the proof of the quadratic growth as in [2].
To verify that the above ingredients imply \(C^{1,1}\) regularity, we split the analysis into two cases. If we are “far” from the free boundary, u locally solves \(\Delta u=g(x,u)\) so by Theorem 1.1 u is \(C^{1,1}\). If we are close to the free boundary then u solves \(\Delta u=g(x,u)\) in a small ball \(B_d(y)\) that touches the free boundary. We invoke (19) for \(0<s<r=d/2\) and the quadratic growth to obtain
for \(s\le d/2\).
So there exists a constant C such that for all \(y \in B_{1/2}\) there exist radii \(r_j(y) \rightarrow 0\) such that
We conclude via Lemma 2.6. \(\square \)
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Acknowledgments
We thank Henrik Shahgholian for introducing us to the regularity problem for semilinear equations. Special thanks go to John Andersson for valuable feedback on a preliminary version of the paper. E. Indrei acknowledges: (i) support from NSF Grants OISE0967140 (PIRE), DMS0405343, and DMS0635983 administered by the Center for Nonlinear Analysis at Carnegie Mellon University and an AMSSimons Travel Grant; (ii) the hospitality of the Max Planck Institute in Leipzig and University of Oxford where part of the research was carried out. L. Nurbekyan was partially supported by KAUST baseline and startup funds and KAUST SRI, Uncertainty Quantification Center in Computational Science and Engineering.
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Communicated by Neil Trudinger.
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Indrei, E., Minne, A. & Nurbekyan, L. Regularity of solutions in semilinear elliptic theory. Bull. Math. Sci. 7, 177–200 (2017). https://doi.org/10.1007/s133730160088z
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DOI: https://doi.org/10.1007/s133730160088z