{'fraction_non_alphanumeric': 0.05417410436298726, 'fraction_numerical': 0.022126281705342688, 'mean_word_length': 3.6986541837331774, 'pattern_counts': {'":': 0, '<': 31, '<?xml version=': 0, '>': 23, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A degenerate fully nonlinear free transmission problem with variable exponents

We study degenerate fully nonlinear free transmission problems, where the degeneracy rate varies in the domain. We prove optimal pointwise regularity depending on the degeneracy rate. Our arguments consist of perturbation methods, relating our problem to a homogeneous, fully nonlinear, uniformly elliptic equation.


Introduction
We consider a free transmission problem where β 0, F is uniformly elliptic and f is bounded and continuous. The model (1) accounts for a diffusion process degenerating as a variable power of the gradient. Transmission problems appear frequently in various fields of physics and biology. They model phenomena which follow different laws in separate subsets of the domain. Typical examples consist of studying mathematical models in composite materials. For a description of these problems, we suggest the readers to the reference [6]. When these subsets depend on the solution itself, these equations become free transmission problems.
Problems of the form belong to a larger class of equations studied in a series of papers by Birindelli and Demengel, starting with the singular case in [2]. The degenerate case was also considered in [3,4]. An important development concerning higher regularity for degenerate fully nonlinear equations was put forward in [12].
In that paper, the authors obtain local C 1,α regularity, for α ∈ (0, α 0 ) and α 1 1 + β , with α 0 corresponding to the C 1,α 0 regularity of the homogeneous equation F D 2 u = 0. In [12,Lemma 6], the authors provide a connection between the homogeneous degenerate equation and the corresponding homogeneous uniformly elliptic equation. This step unlocks a higher regularity class which they access via a tangential path. The methods introduced in [12] resonated, launching new perspectives in the theory of degenerate fully nonlinear equations. In [7], the authors consider the equation where β is allowed to change sign, and obtain local C 1,α regularity, where α ∈ (0, α 0 ) and α with β + and β − corresponding to the positive and negative parts of β, respectively. The estimates obtained in [7] are independent of the continuity modulus of β.
In [11], the authors consider a degeneracy law depending on the sign of the solution. They study the equation which has constant degeneracy rates at each of the phases {±u > 0}, but has a discontinuity across the free boundary ∂{u = 0}. They obtain local C 1,α regularity, for α ∈ (0, α 0 ) and α The authors also establish existence of solutions via Perron's method. Finally, we mention the recent paper [10] where the author considered the following equation where p u (x) = p + χ {u>0} + p − χ {u<0} . In this setting, they prove existence and uniqueness of solutions and obtain local C 1,α regularity.
The present paper consists of two main results. First, we obtain a local regularity result under very general assumptions on β(x, u, Du) which generalizes the results mentioned above. Indeed, we only require β : B 1 × R × R d to be well-defined in its domain and bounded from above and below. The first main result in this paper is the following. Theorem 1 (Local C 1,α regularity). Let u ∈ C(B 1 ) be a viscosity solution to (1). Assume that 0 < β m β(x, t, p) β M for fixed β m , β M ; assume also that F is uniformly (λ, Λ)-elliptic, F (0) = 0 and f is continuous and bounded in B 1 . Let finally α 0 ∈ (0, 1) be given in Remark 3 below. Then, there exist α > 0 and C > 0 such that any viscosity solution u of (1) is in C 1,α (B 1/2 ) and We now state some remarks concerning this result.

Remark 1. Theorem 1 includes the following examples
where θ(t) → 2 as t → 0 and θ(t) → 1 as t → ∞. This equation was considered for the first time in [5].
As one can see from the previous result, there is an intrinsic dependence between the regularity obtained and the degeneracy rate. Hence, if this rate is variable over the domain, it seems natural to obtain regularity results which also vary over the domain. This idea, put forward in Lemma 7 which corresponds to the geometric iterations, is the novelty in the current paper. By considering variable exponents in each iteration, we are able to better capture the pointwise degenerate behaviour of the equation.
To obtain this improved regularity, we consider the following explicit expression for the exponent. Let G i (u, Du) ⊂ B 1 , i = 1, ..., N be disjoint sets which depend on the solution u and its gradient Du, and define Du). Assume the exponent β has the form An example to keep in mind is the following. Let N = 2, If β 0 = 0, β 1 and β 2 are constants, then we recover the result from [11].
Our result is thus more refined, not only in the sense that it includes a much broader class of degeneracies, but also because we obtain an improved pointwise regularity. The second main result in this paper is the following.
Theorem 2 (Pointwise C 1,α regularity). Let u ∈ C(B 1 ) be a viscosity solution to (1). Assume that β is given by assume also that F is uniformly (λ, Λ)-elliptic, F (0) = 0 and f is continuous and bounded in B 1 . Let finally α 0 ∈ (0, 1) be given in Remark 3. Then for every x 0 ∈ B 1/2 , there exist α > 0 and C > 0 such that any viscosity solution u of (1) is pointwise C 1,α (x 0 ) and , Assumptions of the type (3) are typical when obtaining higher regularity of solutions to equations with variable exponents. For example, in [1] the authors are able to prove improved regularity to a class of variational problems with variable exponents, under the assumption above.
Pointwise regularity has been the subject of various papers, see for example [8] and [14]. These are useful when a certain property is not verified locally but instead only at a point. Obtaining such a result as in Theorem 2 instead of a local regularity result as in Theorem 1, comes with a cost, since we must assume stronger uniform continuity of the functions β i . However, more information is gathered. For example, consider N = 0, and assume F is convex, so that α 0 = 1 (see [8,Chapter 6]). Then a local result would yield C 1,α regularity, with α = 1 1001 . The problem with this result is that β 0 ≈ 0 except in a small neighborhood of 0. On the other hand, Theorem 2 would immediately yield C 1,α regularity with α ≈ 1 for points away from the origin.
Another advantage of having such a sharp pointwise regularity comes when studying the free boundary of the problem, where a finer analysis is required.
The remainder of the paper is organized as follows. Section 2 introduces the assumptions to hold throughout the paper, some basic notation and a characterization of Hölder spaces. We also obtain a simple proof for Theorem 1. In Section 3, we simplify the equation, rewriting it as viscosity inequalities and removing the dependence of the exponents on the solution. We then obtain an important smallness assumption, which provides a tangential path between our equation and the homogeneous one. Hölder continuity of a perturbed equation is the topic of Section 4. In Section 5 we derive an approximation lemma. Finally, Section 6 consists of the geometric iterations with variable exponents which combined with the characterization of Hölder spaces put forward in Section 2, provides the improved, pointwise Hölder continuity of the gradient.

Preliminary material and main assumptions
In this introductory section, we present some basic results that will be instrumental in the sequel and detail our main assumptions. We start with some notation.
For r > 0, we call B r (x) the ball in R d centered around x and with radius r. B r denotes B r (0). The space of symmetric d × d real matrices is denoted by S(d). We say u ∈ C 1,α (Ω) if u ∈ C 1 (Ω) and [u] C 1,α (Ω) := sup Similarly, we say u ∈ C 1,α (x 0 ) if u ∈ C 1 in a neighborhood of x 0 and Next, we introduce the uniform ellipticity assumption, assumed to hold throughout the paper.
[A1] (Uniform ellipticity). The operator F : We now make some assumptions on β i . First, assume they have a modulus of continuity which decays at the origin as o(ln(1/t) −1 ).
[A2] (Uniform continuity of the exponents). The exponents β i : Note that [A2] is equivalent to the following statement. For every 0 < r < e −1 , the following holds Hence, by definition, for every ε > 0 there exists δ 1 > 0 such that if ρ δ 1 , then for every k ∈ N, Since ρ will be chosen to be small, we can assume ρ < e −1 , and it follows that Now, by defining ε = α 0 −α 2 (it will become clear later why we make this choice) we also fix δ 1 such that if ρ δ 1 , then for every k ∈ N, This δ 1 depends only on the continuity modulus ω, the universal exponent α 0 introduced in Remark 3, and the exponent α which is defined in Theorem 2.
Remark 4. To emphasize this idea, let's consider some concrete examples.

An example of a modulus of continuity satisfying [A2] is
Finally, we assume that the exponents are bounded uniformly from above and below.
[A3] (Boundedness of the exponents). There exist constants β m and β M such that In the following proposition, we improve slightly the usual proof of Hölder continuity of the gradient to the case with variable exponents.
Proof. Assume without loss of generality that x 0 = 0. The idea is that ℓ k → ℓ uniformly, where ℓ satisfies the desired characterization.
Consider the first order scaling f r (x) := 1 r f (rx). We have, by assumption, since α k+1 − α k 0 and r < 1. First order scaling gives Clearly, this estimate in B 1 implies the following estimates on the coefficients (up to a different K) Since these are Cauchy sequences, we have that Using these estimates, we get where C depends only on r. Now, we apply the usual discretization strategy: for an arbitrary 0 < R < 1, there exists k ∈ N such that r k+1 R < r k . Then, Then r k(α k −α) < C 1 and therefore we get the desired inequality To conclude this introductory section, we present a simple proof of Theorem 1, using the results from [11]. with Then u is a viscosity subsolution to the equation and a viscosity supersolution to the equation Proof. We prove only that if u is a viscosity subsolution to (7), then it is a subsolution to (8), noting that the remaining case follows similarly.
Let ϕ ∈ C 2 (B 1 ) be such that u − ϕ has a local maximum at x 0 . Then Thus, depending on whether |Dϕ(x 0 )| 1 or |Dϕ(x 0 )| < 1 one of the following must hold, respectively provided F D 2 ϕ(x 0 ) 0 (clearly if this is not the case, both inequalities are trivially verified). In either case, we have Hence, we have proved that u is a subsolution of (8).
This simple result places the equation (7) in the framework of [11] with θ 1 = β m and θ 2 = β M (see Proposition 1 therein). A direct application of [11, Theorem 2] yields local regularity u ∈ C 1,α (B 1/2 ) with α = min α − 0 , together with the estimate which implies Theorem 1. The remaining of this paper is devoted to proving Theorem 2. In the next section, we begin our analysis.

Scaling properties
The following result disconnects the dependence of the exponents on the solution, by separating the possible cases.

Lemma 2. Let u ∈ C(B 1 ) be a viscosity solution to the perturbed equation
with β given by (2).
and a viscosity supersolution to the equation Proof. We prove only that if u is a viscosity subsolution to (10), then it is a subsolution to (11), noting that the remaining case follows similarly.
Let ϕ ∈ C 2 (B 1 ) be such that u − ϕ has a local maximum at x 0 . Then We recall that by Theorem 1 we know that u ∈ C 1,α , hence In particular, we have Hence, we have proved that u is a subsolution of (11).
The following result states that to prove Theorem 2, we can assume a smallness regime, without loss of generality. It provides a tangential path between our equation and the homogeneous one.
Proposition 2 (Smallness regime). Let u be a subsolution to the equation and a supersolution to the equation satisfying under the assumption that u L ∞ (B 1 ) 1 and f L ∞ (B 1 ) ε 0 , where C and ε 0 are universal constants. Then, Theorem 2 holds.
Note that we can assume K 1, since otherwise we are already in the smallness regime and we can just take K = 1.
The function u is a viscosity subsolution to min i=0,...,N Similarly, we get max i=0,...,N Since u L ∞ (B 1 ) 1, we note that the smallness assumptions are now satisfied. Hence, if we verify that we can infer that where C 1 depends on ε 0 , which will be fixed universally.
Remark 5. The choice of K in the previous proof differs from the literature (see for example [11]). The observation that K 1 allows us to obtain the simple estimate (15).
In the following section we obtain improved regularity.

Hölder continuity
In this section, we obtain a compactness result for solutions. This result is essential when studying stability since it will allow us to obtain convergence of sequences of solutions.
We start by stating the maximum principle for viscosity solutions, Theorem 3.2 of [9]. Suppose further that (x, y) ∈ Ω × Ω. Then, for every ι > 0, there exist matrices X and Y in S (d) such that

and the matrix inequality
We proceed by stating a result from [13], which we present in the following simplified form.

Proposition 4. Let u ∈ C(B 1 ) be a bounded viscosity subsolution to equation
and a viscosity supersolution to equation Then u ∈ C θ loc (B 1 ) and, for every 0 < τ < 1, there exists C > 0 such that The constant θ depends only on d, λ, Λ and C depends only on d, λ, Λ, γ, u L ∞ (B 1 ) , τ .
Intuitively, in the set where the gradient of a function u is bounded, the function is already Lipschitz. The idea behind the previous result is that if u is a solution of an elliptic equation in the set where its gradient is very large, then we are able to obtain improved regularity.
This proposition will imply Hölder continuity of solutions to (11) and (12) in the case where |p| is sufficiently small. More precisely, let A 0 > 1 (to be fixed) be such that |p| < A 0 . We claim that u is a viscosity subsolution to Indeed, take ϕ ∈ C 2 such that u − ϕ has a local maximum at since we are under the assumption f L ∞ (B 1 ) ε 0 and ε 0 will be chosen very small. Hence, from uniform ellipticity and recalling that F (0) = 0, We verified that u is a viscosity subsolution to (16). In a similar way, we prove that u is a viscosity supersolution to Hence, we proved the following corollary.
ε 0 and assume further that |p| < A 0 . The constants ε 0 and A 0 will be fixed in the sequel.
Then u ∈ C θ loc (B 1 ) for some θ ∈ (0, 1), depending only on d, λ, Λ. In addition, for every 0 < τ < 1, there exists C > 0 such that In the following lemma we obtain Hölder continuity for arbitrary p ∈ R d , which concludes this section.
Then u ∈ C θ loc (B 1 ) for some θ ∈ (0, 1), depending only on d, λ, Λ. In addition, for every 0 < τ < 1, there exists C > 0 such that Proof. We begin by using Proposition 3 to obtain a subjet and a superjet satisfying the estimate (20) below. This was done in [11,Proposition 7] but for completion we replicate the proof.
Fix 0 < r < 1−τ 2 and define For constants L 1 , L 2 > 0 and x 0 ∈ B r , we set We aim at establishing that there exist constants L 1 and L 2 , independent of x 0 , for which L 0. This immediately implies that u is Lipschitz continuous in B τ by taking x 0 = x.
We argue by contradiction. Suppose there exists x 0 ∈ B τ for which L > 0, regardless of the choices of L 1 and L 2 . Consider the auxiliary functions ψ (x, y) .
Let (x, y) be a point where φ attains its maximum. Then Set which implies that x, y ∈ B r (x 0 ). In addition, x = y, since if this isn't the case we would conclude that L 0. We now use Proposition 3 to ensure the existence of a subjet (ξ x , X) of u at x and a superjet (ξ y , Y ) of u at y with Since ω ′ (|x − y|) 1, and for L 1 large enough.
In addition, the matrices X and Y satisfy the inequality for where 0 < ι ≪ 1 depends solely on the norm of Z.
Next we apply the matrix inequality (19) to special vectors as to obtain information about the eigenvalues of X − Y . First, apply it to vectors of the form (z, z) ∈ R 2d to get which implies that all eigenvalues of X −Y are less than or equal to 4L 2 +2ι. Now we apply (19) to the vector z = (σ, −σ) to obtain We thus conclude that at least one eigenvalue of X−Y is below 4L 2 +2ι−4L 1 , which will be a negative number, provided we choose L 1 large enough.
Evaluating the minimal Pucci operator on X − Y , we get for L 1 even larger, if necessary. Furthermore, these jets satisfy the viscosity estimates and max i=0,...,N Since we fixed A 0 = 4L 1 and assumed |p| A 0 , this together with (17) and (18) imply Hence, (21) and (22) imply, respectively, Combining these inequalities with (20) by means of uniform ellipticity, we get which is clearly a contradiction, provided we choose L 1 large enough. This concludes the proof for the case |p| A 0 , which combined with Corollary 1 completes the proof.
With compactness available, we proceed with a key step in our tangential analysis.

Approximation Lemma
We present an approximation lemma for the perturbed equation.

Lemma 4 (Approximation Lemma).
For every 0 < δ < 1, there exists ε 0 > 0 such that, if u ∈ C(B 1 ) is a viscosity subsolution to (11) and a viscosity supersolution to (12) with p = 0, satisfying u L ∞ (B 1 ) 1 and f L ∞ (B 1 ) ε 0 , then one can find a function h which is a viscosity solution to F (D 2 h) = 0 for some F satisfying assumption [A1], such that

Such a function h satisfies
Proof. We argue by contradiction. For simplicity, we split the proof in steps.
Step 1 -Assume that there exist δ 0 > 0 and sequences (u n ) n , (F n ) n and (β n i ) n such that (1) u n ∞ 1; in the viscosity sense, in B 1 ; however, for every function h ∈ C 1,α 0 , it holds Step 2 -Since u n are equibounded in C θ (B 9/10 ), by the Arzelà-Ascoli Theorem they will converge, up to a subsequence, locally uniformly to u ∞ ∈ C(B 1 ). Since F n are (λ, Λ)-elliptic, they are also Lipschitz continuous. Therefore, again by the Arzelà-Ascoli Theorem they will converge locally uniformly to an (λ, Λ)-elliptic operator F ∞ .
Finally, since β n i satisfy assumptions [A2] and [A3] they will converge locally uniformly to continuous functions β ∞ i , respectively. Our goal is to prove that the limiting function u ∞ is a viscosity solution to the equation F ∞ (D 2 u ∞ ) = 0. We only prove that it is a subsolution, since the proof for supersolution is analogous. We will consider two cases, depending on the limit behaviour of (p n ) n .
Step 3: Assume that (p n ) n does not admit a convergent subsequence. Then |p n | → ∞. Let ϕ ∈ C 2 (B 1 ) and assume that u ∞ − ϕ attains a local strict maximum at x 0 ∈ B 1 . By contradiction, assume that There exists a sequence x n → x 0 such that u n − ϕ has a local maximum at x n . Notice that Dϕ(x n ) → Dϕ(x 0 ) and D 2 ϕ(x n ) → D 2 ϕ(x 0 ). Also, by the equation satisfied by u n in the viscosity sense, we have Taking n large enough, we have |Dϕ(x n ) + p n | > 1 and since β n i 0, we get which is inconsistent with (24), when we take the limit n → ∞. Therefore, concluding the proof for the case |p n | → ∞.
Step 4: Suppose now that we can extract a subsequence p n → p ∞ . Resorting to standard stability results (see for example [9, Remarks 6.2 and 6.3]), we conclude that u ∞ is a viscosity subsolution to and a viscosity supersolution to We can assume without loss of generality that p ∞ = 0, i.e., assume that u ∞ is a viscosity subsolution to and a viscosity supersolution to We now claim that these inequalities imply that F ∞ (D 2 u ∞ ) = 0. This is proved in Lemma 5 below.
We present a homogeneous division lemma which concludes the proof of Lemma 4. We follow closely the proof of [12,Lemma 6].

Lemma 5.
Let u ∈ C(B 1 ) be a bounded viscosity subsolution to and a viscosity supersolution to Then u is a viscosity solution to Proof. We prove that (26) implies F D 2 u 0, noting that F D 2 u 0 follows similarly from (25) in a similar way.
If b = 0, then the result is trivial. Hence, assume otherwise. So P (x) = 1 2 x · N x. We argue by contradiction, assuming that F (N ) < 0. By ellipticity, this implies that N has at least one positive eigenvalue. Let S be the subspace generated by the eigenvectors corresponding to the positive eigenvalues and consider the projection P S to this subspace. We consider the following perturbed test function For ǫ large enough, u − ψ attains a negative minimum at B r − B 9 10 r (since S is not empty and u is continuous). Indeed, let m = max Br |u − P | and ǫ = 21m r . Then min x∈B 9 10 r on the other hand, for x ∈ ∂B r , which concludes that the minimum is negative and attained at x 0 ∈ B r \ B 9 10 r . Using this ǫ, we claim that P S x 0 = 0. In fact, since if P S x 0 = 0, we take x = 0 and get But P touches u strictly from below at y = 0 which implies (u − P )(x 0 ) 0 with equality only if x 0 = 0, therefore (u − ψ)(x 0 ) = 0, contradicting the fact that the minimum is negative and that x 0 = 0.
We proved that |P S x 0 | = 0 which implies that ψ is smooth in a neighbourhood of x 0 . Hence, for an appropriate translation of ψ, call itψ, u −ψ has a local minimum in B r at x 0 . Let B be the Hessian of |P S x| at x = x 0 . Note that since |P S x| is a convex function, B 0. We also have the viscosity for e 0 = P S x 0 /|P S x 0 |. Note (N x 0 +ǫe 0 )·P S x 0 > 0, since P S is the projection into the subspace generated by the eigenvalues of N associated with its positive eigenvalues. Then, by ellipticity we obtain which is a contradiction. Hence F D 2 u 0 which concludes the proof.
In the next and final section we provide an iterative scheme to control the oscillation of the gradient.

Hölder continuity of the gradient
In the following lemmas, we proceed with the geometric iteration argument in a sequence of concentric, shrinking balls. The first geometric iteration follows immediately from the approximation lemma.

Lemma 6.
Let u ∈ C(B 1 ) be a viscosity subsolution to (11) and a viscosity supersolution to (12) with p = 0. Under the assumptions of Lemma 4, for every θ < α 0 , there exists a polynomial ℓ(x) = a + b · x and a constant 0 < ρ < 1, depending on ω and universal constants, such that Furthermore, there exists a universal C > 0 such that |a| C and |b| C.
Proof. By Lemma 4, there exists h ∈ C 1,α 0 (B 1/2 ) such that with the uniform estimate This implies that, for every 0 < r ≪ 1 and for the polynomial Let θ < α 0 be arbitrary and take r = ρ given by where δ 1 is given implicitly in (4). Finally fix δ = ρ 1+θ 2 , which also fixes ε 0 via Lemma 4. Then Now we iterate the previous result concentric, shrinking balls.

Lemma 7 (Geometric iterations).
There exist a non-decreasing sequence (α k ) k and universal constants ε 0 > 0 and ρ > 0 such that if u is a viscosity subsolution of (11) and a supersolution of (12) with p = 0, satisfying and Furthermore, the sequence (α k ) k converges to Proof. Assume without loss of generality that x 0 = 0. Take ε 0 and ρ given by Lemma 6, depending on θ, which will be fixed soon. Define the nondecreasing sequence To prove (27) and (28), we will proceed by induction. Let ℓ 0 ≡ 0 and ℓ 1 be given by Lemma 6. Then (27) and (28) hold for k = 1 by Lemma 6.
Assume that (27) and (28) hold up to k. Define Then v k L ∞ (B 1 ) 1 and v k is a viscosity subsolution to min i=0,...,N This implies the following estimate min i=0,...,N where the last inequality follows from the definition of α k . Calling p k := ρ −kα k b k and β k i (x) := β i (ρ k x), we get that v k is a subsolution to min i=0,...,N Similarly, we prove that v k is a viscosity supersolution to max i=0,...,N Note that β k i still satisfy assumption [A2].
Hence, we can use Lemma 6 to guarantee the existence of a linear function ℓ(x) = a + p · x such that sup Bρ |v k − ℓ| ρ 1+θ , where θ = α+α 0 2 < α 0 and the coefficients satisfy a = h(0), where h is a viscosity solution to G(D 2 h) = 0 and G has the same ellipticity constants as F . Hence, as a straightforward application of ellipticity, h has interior C 1,α 0 estimates which imply universal bounds on the coefficients of ℓ.