Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian © The Author(s) 2022 Published version Blanc, Pablo; Han, Jeongmin; Parviainen, Mikko; Ruosteenoja, Eero Blanc, P., Han, J., Parviainen, M., & Ruosteenoja, E. (2022). Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian. Potential Analysis, Early online. https://doi.org/10.1007/s11118-022-10037-6 2022 Potential Analysis https://doi.org/10.1007/s11118-022-10037-6 Game-Theoretic Approach to H¨ older Regularity for PDEs Involving Eigenvalues of the Hessian Pablo Blanc1·Jeongmin Han1·Mikko Parviainen1·Eero Ruosteenoja1 Received: 7 January 2022 / Accepted: 6 July 2022 ©The Author(s) 2022 Abstract WeprovealocalH ¨ older estimate for any exponent 0 <δ<1 2for solutions of the dynamic programming principle uε(x) = n  j=1 αjinf dim(S)=jsup v∈S |v|=1 uε(x +εv) +uε(x −εv) 2 with α1,α n>0andα2,···,α n−1≥0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE n  i=1 αiλi(D2u) =0, where λ1(D2u) ≤···≤λn(D2u) are the eigenvalues of the Hessian. Keywords Dynamic programming principle ·H¨ older estimate ·Viscosity solution · Eigenvalue of the Hessian ·Fully nonlinear PDEs Mathematics Subject Classification (2010) 91A05 ·91A15 ·35D40 ·35B65 Jeongmin Han [email protected] Pablo Blanc [email protected] Mikko Parviainen [email protected] Eero Ruosteenoja [email protected] 1Department of Mathematics and Statistics, University of Jyv¨ askyl¨ a, P.O. Box 35, FI-40014 Jyv¨ askyl¨ a, Finland P. Blanc et al. 1 Introduction 1.1 Main results In this paper, we show local H¨ older regularity for solutions of the following Dynamic Programming Principle (DPP) uε(x) = n  j=1 αjinf dim(S)=jsup v∈S |v|=1 uε(x +εv) +uε(x −εv) 2(1.1) in a bounded domain ⊂Rn,whereα1,...,α n≥0, min(α1,α n)>0, and n j=1αj=1. Our main result is the following, restated as Theorem 4.1. Main Theorem Let uεbe a function satisfying the DPP (1.1)in a bounded domain .Then for any 0<δ<1 2and x,z ∈Brwith B2r⊂, there exists a constant C=C(δ,α1,α n)> 0such that |uε(x) −uε(z)|≤C||uε||L∞(B2r)|x−z|δ rδ+εδ rδ. (1.2) That the above theorem holds for any 0 <δ<1 2is explicitly obtained in the proof in Eq. 3.14. The DPP (1.1) has a connection to a certain PDE involving eigenvalues of the Hessian. Indeed, under certain regularity assumptions for the boundary of the domain, when ε→0, solutions of Eq. 1.1 converge uniformly to the unique viscosity solution of the following PDE, n  i=1 αiλi(D2u) =0,(1.3) where λ1(X) ≤···≤λn(X) are the ordered eigenvalues of X∈S(n),thesetofn×nreal symmetric matrices. As a consequence of our main result, we obtain the same H¨ older estimate for any 0 < δ<1 2for viscosity solutions of this PDE. For the proof of the following corollary, see Section 2.3. Corollary 1.1 Let ube the viscosity solution of Eq. 1.3 in a bounded domain . Then for any 0<δ<1 2and x,z ∈Brwith B2r⊂, there exists a constant C=C(δ,α1,α n)>0 such that |u(x) −u(z)|≤C||u||L∞(B2r)|x−z|δ rδ. We remark that n i=1αiλi=0 satisfies Pucci type inequalities, and thus we can get H¨ older regularity from the general theory directly, even though the exponent is not explicitly given (see the beginning of Section 4). Game-Theoretic Approach to H¨ older Regularity... For this equation with lower order terms, Ferrari and Vitolo [11] used methods from the viscosity theory to study ABP, Harnack and H¨ older estimates, and later Ferrari and Galise [10]showedC0,δ-regularity for δ=1−α1+αn (√α1+√αn)2∈(0,1 2]. We note that δ=1 2holds if and only if α1=αn. Also observe that min(α1,α n)>0 is necessary in [11]aswellasin our main result. The DPPs of type (1.1) model competition of two players, and one can relate these games to different applications. For example, in the context of related tug-of-war games, they have been suggested in connection to the option pricing problem with market manipulation [17]. To be more precise, for a given boundary data, the solution of the DPP (1.1)isalsothe value function of a two-player zero-sum stochastic game, the rules of which can be read from the DPP. We will describe the game in more detail in the next section, but informally, a token is placed at x∈,αjis the probability that the number jis chosen. Then the player aiming to minimize the value chooses a j-dimensional subspace of Rn, and finally, the player aiming to maximize the value chooses a unit vector from that subspace. Then the token is moved an ε-step either to the direction or the opposite direction of the vector, with equal probabilities. The game continues until the token is moved outside of ,andthe player choosing subspaces pays the amount given by the boundary payoff function to the other player. To the best of our knowledge, there are no prior works studying local regularity of DPPs or games related to fully nonlinear PDEs involving eigenvalues of the Hessian. The game that we just described is connected to the PDE (1.3). This connection has been studied in detail by Blanc and Rossi [5]forthePDEλj(D2u) =0, where j∈{1,...,n}. See also [7] for a game associated to the Dominative p-Laplacian and [3,13]forgames associated to parabolic versions of these equations. The rest of the paper is organized as follows. In the following subsection, we give a more detailed idea of our proof method. In Section 2we give some preliminary definitions and results for viscosity solutions. In Section 3we prove the main result for the special case α1=αn=1 2. In Section 4we prove the main theorem. 1.2 Method of the Proof Although the ideas behind the proof of our main theorem stem from games, we do not use methods from stochastic game theory. Instead, our starting point is the coupling method introduced by Luiro and Parviainen [15] in the context of tug-of-war games. However, a direct application of their method does not seem to work in our case, so we need a new type of coupling, which is the main novelty of this work. To give an idea of the proof, for simplicity we will discuss a special case α1=αn=1 2, in which case the DPP (1.1) can be written as uε(x) =1 2sup |v|=1uε(x +εv) +uε(x −εv) 2+1 2inf |w|=1uε(x +εw) +uε(x −εw) 2. The starting point of the coupling is to define a 2n-dimensional game related to the DPP. Notice that the function g:×→Rgiven by g(x,z) =uε(x) −uε(z) P. Blanc et al. can be written as a solution of a suitable DPP in R2nas follows, g(x,z) =uε(x) −uε(z) =1 2sup |v|=1uε(x +εv) +uε(x −εv) 2+1 2inf |˜v|=1uε(x +ε˜v) +uε(x −ε˜v) 2 −1 2sup |˜w|=1uε(z +ε˜w) +uε(z −ε˜w) 2−1 2inf |w|=1uε(z +εw) +uε(z −εw) 2 =1 2sup |v|=1 |w|=1uε(x +εv) +uε(x −εv) −uε(z +εw) −uε(z −εw) 2 +1 2inf |˜v|=1 |˜w|=1uε(x +ε˜v) +uε(x −ε˜v) −uε(z +ε˜w) −uε(z −ε˜w) 2. The potential of this DPP is to transform the question of regularity of uεto the question of the absolute size of g. The heuristic idea is to introduce a suitable stochastic game in ×, where we aim to move the two tokens to the diagonal set T:= {(x, z) ∈×:x=z}, where g=0, before our opponent can move the tokens outside of the set ×.Observe that uε(x +εv) +uε(x −εv) −uε(z +εw) −uε(z −εw) 2 =g((x, z) +ε(v, w)) +g((x, z) −ε(v, w)) 2. Following the idea of Luiro and Parviainen, we could consider a 2n-dimensional game where each player (both with probability 1 2) gets to choose vand wand then the tokens move to (x, z) +ε(v, w) or to (x, z) −ε(v, w), each possibility with probability one half. If we set the boundary values of our game to be 0 on Tand 2 sup uεin R2n\(×) and could prove that |g(x, z)|≤C|x−z|δ, we would get a desired H¨ older estimate for the function uε. Unfortunately, these rules for the 2n-dimensional game do not seem to be suitable to obtain regularity estimates in our case. The problem is that our opponent can force the tokens away by choosing w=−vnormal to x−z. Observe that if the new position of the tokens is given by (˜x, ˜z) =(x, y) +ε(v, w),weget|˜x−˜z|2=|x−z|2+4ε2.Thesame holds for (˜x, ˜z) =(x, y) +ε(−v,−w). Observe that it also holds uε(x +εv) +uε(x −εv) −uε(z +εw) −uε(z −εw) 2 =g((x, z) +ε(v, −w)) +g((x, z) −ε(v, −w)) 2. Again the rules that follow from this formula allow our opponent to force the tokens away from each other, in this case by choosing w=vnormal to x−z. In conclusion, with the rules that we have described, we do not get a suitable coupling. Our new idea is to couple the moves in one way or the other depending on the choice of the vectors vand w. When the tokens are placed at xand z,givenvand wwe define two rules moving the token: (i) the token moves to (x, z) +ε(v, w) or to (x, z) −ε(v, w), each possibility with probability one half. Game-Theoretic Approach to H¨ older Regularity... (ii) the token moves to (x, z) +ε(v, −w) or to (x, z) −ε(v, −w), each possibility with probability one half. Let us define the 2n-dimensional game. We toss a coin and the winner of the toss chooses two unitary vectors vand w.Sety=x−z. We also write vy⊥=v−v,y y,yyand wy⊥= w−w,y y,yy.If|vy⊥|2+|wy⊥|2>1andvy⊥,w y⊥<0, then the tokens move according to rule (ii). In any other case the tokens move according rule (i). Define F(x,z,v,w,g) =g((x,z)+ε(v,−w))+g((x,z)−ε(v,−w)) 2if |vy⊥|2+|wy⊥|2>1andvy⊥,w y⊥<0, g((x,z)+ε(v,w))+g((x,z)−ε(v,w)) 2otherwise. (1.4) Then we obtain the DPP for our 2n-dimensional game g(x,z) =1 2sup |v|=1 |w|=1 F(x,z,v,w,g)+1 2inf |v|=1 |w|=1 F(x,z,v,w,g). (1.5) Observe that in the case of Fig. 1A, that is when a player selects w=−vnormal to x−z, we have to move the tokens accordingly to rule (ii). We have x+εv −(z −εw)) = x−εv −(z +εw)) =x−z, and therefore the distance between the tokens is preserved. Now, we consider the case in Fig. 1B, that is, vand ware normal to x−zandalsoto each other. If the new position of the tokens is given by (˜x, ˜z) =(x, z) +ε(v, w),weget |˜x−˜z|2=|x−z|2+2ε2.Wegetthesameif(˜x, ˜z) =(x, z) +ε(v, −w). Then, the tokens are forced away from each other independently of what rule we select. But still, observe that this growth is smaller than the one we were getting in the case of Fig. 1A when applying rule (i), since |˜x−˜z|2=|x−z|2+4ε2in that case. We claim that our choice of when to apply (i) or (ii) reduces the ability of the players to push the tokens away from each other, and this is the key of the matter. 2 Preliminaries In this section, we include some preliminary results concerning solutions to the equation and the game. Fig. 1 Two choices of vectors when the tokens are at (x, z) P. Blanc et al. 2.1 Viscosity Solutions Let us begin by recalling the definition of viscosity solutions for Eq. 1.3. We denote by USC() (resp. LSC()) the set of all upper (resp. lower) semicontinuous functions defined on . Definition 2.1 Let u:¯ →Rbe a function. (a) We say that u∈USC() is a viscosity subsolution to Eq. 1.3 if for all x∈and φ∈C2() such that u(x) =φ(x) and u(y) < φ(y) for y= xwe have n  i=1 αiλi(D2φ) ≥0. (b) We say that u∈LSC() is a viscosity supersolution to Eq. 1.3 if for all x∈and φ∈C2() such that u(x) =φ(x) and u(y) > φ(y) for y= xwe have n  i=1 αiλi(D2φ) ≤0. If uis continuous and satisfies both of (a) and (b), we say that uis a viscosity solution to Eq. 1.3. Next, we prove comparison and thus uniqueness to our operator. It would follow from [2], but here we give a simple alternative proof for this particular operator. Remark 2.2 If Mis a Hermitian matrix, it is diagonalizable with real eigenvalues and by the Min-max Theorem those eigenvalues verify λj(M) =inf dim(S)=jsup |v|=1Mv,v for j=1,...,N, and we can use this identity for the Hessian matrix. Theorem 2.3 Let u1∈USC() be a viscosity subsolution (1.3), and u2∈LSC() a viscosity supersolution to Eq. 1.3. If u1≤u2on ∂,thenu1≤u2on . Proof First, we make a counter proposition sup  (u1−u2)=: θ>0. Then observe that the equation can be written as n  i=1 αiλi(D2u1)= n  i=1 αiinf dim(S)=jsup |v|=1D2u1(x)v, v. Moreover, ˜u1(x) =u1(x) +δ|x|2is a subsolution to n  i=1 αiinf dim(S)=jsup |v|=1D2˜u1(x)v, v= n  i=1 αiinf dim(S)=jsup |v|=1D2(u1(x) +δ|x|2)v, v =2Cδ. Game-Theoretic Approach to H¨ older Regularity... Let (x,y) =˜u1(x) −u2(y) −ϕ(x,y) := ˜u1(x) −u2(y) −1 2ε|x−y|2, and let (xε,y ε)be the maximum point on ×. The points xε,y εare not at the boundary in a bounded domain for small enough δby the standard theory. Then by the theorem of sums [9]wehave (Dxϕ(xε,y ε), X) ∈J2,+˜u1(xε), (−Dyϕ(xε,y ε), Y ) ∈J2,−u2(yε), and X≤Y. Furthermore, let η>0, Sjbe a j-dimensional subspace of Rnand vj∈Sj with vj=1 be such that sup v∈Sj,|v|=1Yv,v≤ inf dim(S)=jsup |v|=1Yv,v+η and η+Xvj,v j≥ sup v∈Sj,|v|=1Xv, v. Then we have 2Cδ ≤ n  i=1 αiinf dim(S)=jsup |v|=1Xv, v− inf dim(S)=jsup |v|=1Yv,v ≤ n  i=1 αiinf dim(S)=jsup |v|=1Xv, v− sup v∈Sj,|v|=1Yv,v+η ≤ n  i=1 αisup v∈Sj,|v|=1Xv, v− sup v∈Sj,|v|=1Yv,v+η ≤ n  i=1 αiXvj,v j− sup v∈Sj,|v|=1Yv,v+2η ≤ n  i=1 αi(Xvj,v j−Yv j,v j)+2η ≤ n  i=1 αi(X −Y)v j,v j+2η≤2η, which is a contradiction for small enough η>0. It might also be instructive to think some special cases. For example, for the first eigenvalue equation λ1(D2u)(x) := inf|v|=1D2u(x)v, v=0, choosing Yv 0,v 0= inf|v|=1Yv,v, the key computation above reads as 2δ≤inf |v|=1Xv, v− inf |v|=1Yv,v ≤inf |v|=1Xv, v−Yv 0,v 0+η ≤Xv0,v 0−Yv 0,v 0+η ≤(X −Y)v 0,v 0+η≤η, a contradiction. P. Blanc et al. Observe that uniqueness immediately follows from the comparison principle for the viscosity solutions of the boundary value problem n i=1αiλi(D2u) =0in, u=gon ∂, with given continuous boundary values g:∂ →R. Also, observe that if the domain is strictly convex, we have that a plane can act as a barrier. Then, the solution obtained by Perron’s method turns out to be continuous up to the boundary. A weaker condition for the existence of continuous solutions for smooth domains can be found in [12]. 2.2 Games A game associated with the equation λj(D2u) =0 was introduced in [5]. Here we modify the game so that it is associated with Eq. 1.3. Next, we give the precise formulation of the game. It is a two-player zero-sum game. Fix a domain ⊂RN,ε>0 and a final payoff function G:RN\→ R. The rules of the game are the following: the game starts with a token at an initial position x0∈and develops in several rounds. At the beginning of each round j∈{1,...,n}is chosen at random such that P(j =i) =αifor each i=1,...,n. With this given value, Player I chooses a subspace Sof dimension jand subsequently Player II a unitary vector v∈S. Then the position of the token is moved to x±εv with equal probabilities. The game continues until the token leaves the domain. At this time that we call τ,PlayerIpaysG(xτ)to Player II. When the two players fix their strategies SIand SII, we can compute the expected outcome as Ex0 SI,SII [G(xτ)]. Then the value of the game for any x0∈is defined as uε(x0)=sup SI inf SII Ex0 SI,SII [G(xτ)]=inf SII sup SI Ex0 SI,SII [G(xτ)], and verifies the DPP (1.1), that is uε(x) = n  j=1 αjinf dim(S)=jsup v∈S |v|=1 uε(x +εv) +uε(x −εv) 2(2.1) for x∈and uε(x) =G(x) for x∈ ,see[4]. Intuitively, the rules of the game can be seen from the DPP. When Player I, who aims to minimize the value, chooses a subspace, she knows that Player II aims to choose from that subspace a unitary vector maximizing the average ‘ε-step value’. 2.3 Application to the PDE (1.3) We give a brief explanation of the relation between Eqs. 1.1 and 1.3, and how to prove Corollary 1.1 using the result of our main theorem. If we assume that the domain is strictly convex, we obtain that uε→u(2.2) uniformly as ε→0whereuis the unique solution to Eq. 1.3. Observe that in [5] a condition over the boundary is given for each j. This condition is used to prove that the game value is asymptotic continuous near the boundary. It is in this step that we use that the domain to Game-Theoretic Approach to H¨ older Regularity... 4 The General Case In this section, we consider the DPP (1.1) related to the PDE (1.3). We rewrite the equation and present the rules of the game in a slightly different way. We assume α= 2min{α1,α n}>0. We define β=1−α,βi=αi/β for i=2,...,n −1and βi=(αi−α/2)/β for i=1,n. We can rewrite Eq. 1.3 as αλ1+λn 2+β n  i=1 βiλi=0. (4.1) We remark that one can derive H¨ older regularity for Eq. 4.1, since a viscosity solution to Eq. 4.1 satisfies 1−α 2 λi<0 λi+α 2n λi>0 λi≤1−(n −1)α 2nλ1+α 2n n  i=2 λi ≤αλ1+λn 2+β n  i=1 βiλi ≤1−(n −1)α 2nλn+α 2n n−1  i=1 λi ≤1−α 2 λi>0 λi+α 2n λi<0 λi. These Pucci type inequalities are what is required to use [8, Proposition 4.10]. Now the game for αican be presented in the following way: at every round with probability αwe play the game for 1 2λ1+1 2λnand with probability βwe play the game according to βi. In this case, the related DPP is uε(x) =α 2sup |v|=1 uε(x +εv) +uε(x −εv) 2+α 2inf |w|=1 uε(x +εw) +uε(x −εw) 2 +β n  i=1 βiinf dim(S)=jsup v∈S |v|=1 uε(x +εv) +uε(x −εv) 2. (4.2) This is equivalent to Eq. 1.1. In order to define the 2n-dimensional game related to Eq. 4.2, first we define a 2ndimensional game related to λj.Fixj∈{1,...,n}. We consider the game related to λjand write ujfor its value function. It satisfies the following DPP uj(x) =inf dim(S)=jsup v∈S |v|=1 uj(x +εv) +uj(x −εv) 2 for any x∈. P. Blanc et al. Set gj(x, z) =uj(x) −uj(z).Wehave inf dim(S)=jsup v∈S |v|=1 uj(x +εv) +uj(x −εv) 2−inf dim(˜ S)=j sup ˜v∈˜ S |˜v|=1 uj(z +ε˜v) +uj(z −ε˜v) 2 =sup dim(˜ S)=j inf dim(S)=jsup v∈S |v|=1 inf ˜v∈˜ S |˜v|=1uj(x+εv)+uj(x−εv) 2−uj(z+ε˜v)+uj(z−ε˜v) 2 =sup dim(˜ S)=j inf dim(S)=jsup v∈S |v|=1 inf ˜v∈˜ S |˜v|=1 gj(x +εv, z +ε˜v) +gj(x −εv, z −ε˜v) 2. We can read the rules of the 2n-dimensional game as follows: Player II selects ˜ S,PlayerI the subspace Sand then Player II a unitary vector v∈Sand Player I a vector ˜v∈˜ S. Then, the token moves to (x, z) ±ε(v, ˜v) each with probability one half. Combining the above observation for each gjwith the 2n-dimensional DPP for the game associated with 1 2λ1+1 2λn, for the function g(x, z) =uε(x) −uε(z),wehave g(x,z) =α 2sup |v|=1 |w|=1 F(x,z,v,w,g)+α 2inf |˜v|=1 |˜w|=1 F(x,z,v,w,g) +β n  i=1 βisup dim(˜ S)=j inf dim(S)=jsup v∈S |v|=1 inf ˜v∈˜ S |˜v|=1 g(x +εv, z +ε˜v) +g(x −εv, z −ε˜v) 2, where Fis the function given by Eq. 1.4. Now we state and prove the H¨ older regularity result for Eq. 1.1. Theorem 4.1 Let uεbe a function satisfying the DPP (1.1)in a bounded domain .Then for any 0<δ<1 2and x,z ∈Brwith B2r⊂, there exists a constant C=C(δ,α) > 0 such that |uε(x) −uε(z)|≤C||uε||L∞(B2r)|x−z|δ rδ+εδ rδ. Proof Recall the barrier function fin the proof of Theorem 3.1. By a similar argument as in the previous section, it is enough to show that f(x,z) > α 2sup |v|=1 |w|=1 F(x,z,v,w,f)+α 2inf |v|=1 |w|=1 F(x,z,v,w,f) +β n  i=1 βisup dim(˜ S)=j inf dim(S)=jsup v∈S |v|=1 inf ˜v∈˜ S |˜v|=1 f(x+εv, z +ε˜v) +f(x−εv, z −ε˜v) 2. (4.3) We first consider the case |x−z|>N 10 ε. For the terms involved in the game associated with 1 2λ1+1 2λnwe can recall the estimate Eq. 3.15. Meanwhile, for the game associated to Game-Theoretic Approach to H¨ older Regularity... λj, we observe that by taking S=˜ Sand ˜v=vwe get sup dim(˜ S)=j inf dim(S)=jsup v∈S |v|=1 inf ˜v∈˜ S |˜v|=1 f(x+εv, z +ε˜v) +f(x−εv, z −ε˜v) 2 ≤sup dim(˜ S)=j sup ˜v∈S |˜v|=1 f(x+ε˜v,z +ε˜v) +f(x−ε˜v,z −ε˜v) 2. (4.4) Moreover, we observe that sup dim(˜ S)=j sup ˜v∈S |˜v|=1 f(x+ε˜v,z +ε˜v) +f(x−ε˜v,z −ε˜v) 2=f(x,z)+4ε2. (4.5) We conclude that α 2sup |v|=1 |w|=1 F(x,z,v,w,f)+α 2inf |˜v|=1 |˜w|=1 F(x,z,v,w,f) +β n  i=1 βisup dim(˜ S)=j inf dim(S)=jsup v∈S |v|=1 inf ˜v∈˜ S |˜v|=1 f(x+εv, z +ε˜v) +f(x−εv, z −ε˜v) 2 <f(x,z)−α˜ Cε2+4βε2, where ˜ Cis the constant in Eq. 3.15.Thus,ifwetake ˜ Clarge enough such that −˜ Cα +4β<0, we obtain Eq. 4.3. Next we assume that |x−z|≤N 10 ε.FromEqs.4.3,4.4 and 4.5, it is enough to show that f(x,z)> α 2sup |v|=1 |w|=1 F(x,z,v,w,f)+α 2inf |v|=1 |w|=1 F(x,z,v,w,f)+β n  i=1 βi(f (x, z) +4ε2). This can be rewritten as f(x,z)> 1 2sup |v|=1 |w|=1 F(x,z,v,w,f)+1 2inf |v|=1 |w|=1 F(x,z,v,w,f)+4(1−α) αε2. (4.6) We use a similar argument in the proof of Theorem 3.1, but we choose Csufficiently large such that sup |v|=1 |w|=1 F(x,z,v,w,f 2)≥8(1−α) α+6Cεδ+2f2(x, z) in Eq. 3.16.ThenwegetEq.4.6, which completes the proof. 4.1 The Dominative p -Laplacian Recently, there has been some interest to the Dominative p-Laplacian Dpu:= λ1+···+λn−1+(p −1)λn, where 2 ≤p<∞,see[6]. It explains a superposition of p-superharmonic functions, which was studied in [14]. P. Blanc et al. The dominative p-Laplacian can be regarded as a special case of the operator n i=1αiλi, which has been considered so far. Observe that the equation Dpu=0 is equivalent to the equation (1.3)whenαi=1 n+p−2for i=1,···,n−1, and αn=p−1 n+p−2. Therefore, by plugging these values in Eq. 1.1 we obtain the following DPP uε(x) =1 n+p−2 n−1  j=1 inf dim(S)=jsup v∈S |v|=1 uε(x +εv) +uε(x −εv) 2 +p−1 n+p−2sup v∈S |v|=1 uε(x +εv) +uε(x −εv) 2. Since min{α1,α n}>0, our result, Theorem 4.2, covers the solutions to this DPP. We also remark here that the operator Dpis uniformly elliptic, and thus we can obtain C1,δ-regularity for viscosity solutions of the equation Dpu=0(see[8, Section 5.3]). A different game associated to the Dominative p-Laplacian was presented in [7](see also [13]). Their DPP reads as follows uε(x) =qBε(x) uε(y)dy +(1−q) sup |v|=1uε(x +εv) +uε(x −εv) 2,(4.7) where q=n+2 n+p. This is a control problem. Let x0be the starting point. The player first chooses a unit vector v, and then the token is moved according to the following rules: x1 is randomly selected in Bε(x0)with probability n+2 n+p,andx1=x0±εv with probability p−2 2(n+p) , respectively. This stochastic process is repeated until the token leaves . The player tries to maximize the expected value of G(xτ), and thus he/she chooses the direction vfor this purpose. We can also obtain the following regularity result for this DPP with a slightly worse upper bound for δ. Theorem 4.2 Let uεbe a function satisfying the DPP (4.7)in a bounded domain .Then for any 0<δ< 1 10 and x,z ∈Brwith B2r⊂, there exists a constant C=C(δ) > 0 such that |uε(x) −uε(z)|≤C||uε||L∞(B2r)|x−z|δ rδ+εδ rδ. Proof We first observe that sup |v|=1uε(x +εv) +uε(x −εv) 2−sup |w|=1uε(z +εw) +uε(z −εw) 2 =sup |v|=1 inf |w|=1uε(x +εv) +uε(x −εv) −uε(z +εw) −uε(z −εw) 2. Like before, we again consider g(x,z) =uε(x) −uε(z). Game-Theoretic Approach to H¨ older Regularity... Then we have g(x,z) =uε(x) −uε(z) =qBε(x) uε(y)dy −Bε(z) uε(y)dy +(1−q) sup |v|=1 inf |w|=1uε(x +εv) +uε(x −εv) −uε(z +εw) −uε(z −εw) 2 =qBε(x) uε(y)dy −Bε(z) uε(y)dy +(1−q) sup |v|=1 inf |w|=1g((x, z) +ε(v, w)) +g((x, z) −ε(v, w)) 2 ≤qBε(x) uε(y)dy −Bε(z) uε(y)dy +(1−q) sup |v|=1g((x, z) +ε(v, v)) +g((x, z) −ε(v, v)) 2. Again, we recall the auxiliary function fand the ideas explained before. In [15, Section 4], we can find the following observation Bε(x) uε(y)dy −Bε(z) uε(y)dy =1 |Bε|Bε(0)\Bε(z−x) u(x+h) −u(z +Px,z(h))dh, where Px,z is a map to send a point to its mirror point with respect to span(x −z)⊥,the orthogonal complement of the subspace generated by x−z. Repeating a similar calculation in the proof of Theorem 3.1, we see that it is enough to deduce a contradiction to Eq. 3.5 if we prove f(x,z) > q·1 |Bε|Bε(0)\Bε(z−x) f(x+h, z +Px,z(h))dh +Bε(x)∩Bε(z) f(y,y)dy +(1−q) sup |v|=1f ((x, z) +ε(v,v)) +f ((x, z) −ε(v,v)) 2(4.8) for every (x.z) ∈B1×B1(see also (4.25) in [15]). Assume δ<1/10 and set C=1010 δ2ω,whereωis to be determined. We refer to the following estimate in [15, Section 4]: 1 |Bε|Bε(0)\Bε(z−x) f(x+h, z +Px,z(h))dh +Bε(x)∩Bε(z) f(y,y)dy−f(x,z) <Kε δ, P. Blanc et al. where K=|x−z|δ−210 −Cδ 4(n+2)if |x−z|>Nε/10, −C2 4n+3C+1if |x−z|≤Nε/10. Thus, by choosing ω≤4−n,weget 1 |Bε|Bε(0)\Bε(z−x) f(x+h, z +Px,z(h))dh +Bε(x)∩Bε(z) f(y,y)dy−f(x,z) <−˜ Cεδ(4.9) for ˜ C=min{1 4(Cδ 4(n+2)−10), C2 4n−3C−1}. Nothe that ˜ Cis strictly positive because C=1010 δ2ω. We also observe that sup |v|=1f ((x, z) +ε(v,v)) +f ((x, z) −ε(v, v)) 2=f(x,z)+4ε2 for any (x, z) ∈B1×B1. Combining this with Eq. 4.9,wehave q·1 |Bε|Bε(0)\Bε(z−x) f(x+h, z +Px,z(h))dh +Bε(x)∩Bε(z) f(y,y)dy +(1−q) sup |v|=1f ((x, z) +ε(v,v)) +f ((x, z) −ε(v, v)) 2−f(x,z) <−q˜ Cεδ+4(1−q)ε2 <−q˜ C+4(1−q)εδ. Now we can complete the proof if we choose ωsmall such that −q˜ C+4(1−q) < 0. Combining the above estimates, we obtain Eq. 4.8. Acknowledgements The authors would like to thank Julio D. Rossi for useful discussions. Funding Open Access funding provided by University of Jyv¨ asky¨ a (JYU). J. H. was supported by NRF2021R1A6A3A14045195. P. B. and M. P. were partly supported by the Academy of Finland project 298641. Data Availability This manuscript has no associated data. Declarations Competing interests The authors declare that there is no conflict of interest regarding the publication of this article. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Game-Theoretic Approach to H¨ older Regularity... References 1. 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