Additive Eigenvalue Problems of the Laplace Operator with the Prescribed Contact Angle Boundary Condition

Additive eigenvalue problem appears in ergodic optimal control or the homogenization of Hamilton–Jacobi equations. It has wide applications in several fields including computer science and then attracts the attention. In this paper, we consider the Poisson equations with the prescribed contact angle boundary condition and finally derive the existence and the uniqueness of the solution to the additive problem of the Laplace operator with the prescribed contact angle boundary condition.


Introduction
Additive eigenvalue problem appears in ergodic optimal control or the homogenization of Hamilton-Jacobi equations. In ergodic optimal control, the additive eigenvalue corresponds to averaged long-run optimal costs, while it determines the effective Hamiltonian in the homogenization of Hamilton-Jacobi equations. It is usually applied to study the large time behavior of the Cauchy problem of Hamilton-Jacobi equations. As a character of large time behavior, it also appears in the fields of computer science, big datadriven cloud service recommendation, etc. One can refer to the related references such as [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. In pure mathematics, it has been studied by so many mathematicians, such as Lions [16] and Ishii [17]. More introduction can be found in [17] and the references therein.
e Poisson equation is a kind of simple but interesting and important object in the field of partial differential equations, and it appears in lots of mathematical modelling related to the real world. Also, it is always submitted with several kinds of boundary conditions. Among various boundary conditions, Gilbarg and Trudinger boundary and Neumann boundary conditions are already included in the classical theory, and one can refer to the study [18]. For the capillary-type boundary conditions, there are relatively less results. In [19], Xu derived the gradient estimate for Poisson equations with the prescribed contact angle boundary condition. In this paper, we consider the additive eigenvalue problem, and our main result is related to the additive eigenvalue problem of the Laplace operator with the prescribed contact angle boundary condition.

Theorem 1.
Let Ω be a bounded domain in R n with a smooth boundary, n ≥ 2, then ] is the inner normal vector of zΩ . For any θ ∈ C ∞ (Ω), and |cos θ| ≤ b < 1, there exist a unique τ ∈ R and a smooth function w ∈ C ∞ (Ω) solving Moreover, the solution to (1) is unique up to a constant. Our work is motivated by studies [20][21][22]. In [20], Arisawa got the classical solution to a kind of fully nonlinear uniformly elliptic equation with oblique derivative boundary conditions which comes from the stochastic control fields. In [21], Francesca studied the large time behavior as t ⟶ + ∞ of the viscosity solution χ to a Neumann boundary value problem where F and L are at least continuous functions defined, respectively, on Ω × R n × S n and Ω × R n and got a con- under the assumption that F and L satisfy several structure conditions. Especially, we remark that the condition is obviously not satisfied by the prescribed contact angle boundary value. Also, it is a bit of regret that the study [21] only discussed the viscosity solution and one may expect for the classical solution even if some more strong conditions are imposed on F. In [22], Gao et al. adopted a blow-up technique to conclude the a priori estimate to and then in [23], Huang and Ye concluded the large time behavior of the solution in the classical senses. ey proved that the solution will converge to one of the following additive eigenvalue problems with the Neumann boundary condition as the time goes to infinity: A similar convergence result was obtained by Ma et al. [24] for the graphic mean curvature flow with the Neumann boundary condition: where Ω is a strictly convex bounded domain in R n with the smooth boundary for n ≥ 2 and u 0 (x) and φ(x) are smooth functions satisfying u 0,] � φ(x) on zΩ. ey proved that, up to a constant, the solutions converge to a translating solution λt + w. In other words, (w, λ) is a solution to n i,j�1 In fact, the result in [24] arose from the prescribed contact angle boundary condition for mean curvature-type equations which is the more complicated case, and till now, it is an open problem to get the translating soliton for the graphic mean curvature flow with the prescribed contact angle boundary condition except for the two-dimension case and for the free boundary case. For more details, one can refer to studies [25,26], etc.
When we try to prove eorem 1, we actually can follow the procedure in [24] to get the uniform gradient estimate without the C 0 a priori estimate. But for the Laplace case, it seems to be unnatural to require the strict convexity condition of the domain and we have to follow the blow-up technique adopted in [20,22].
We firstly give some notations.
en, there exists a positive constant σ 1 > 0 such that As mentioned by Lieberman [27] or Simon and Spruck [28], we can extend ] as Dd in Ω σ which is a C 2 vector field. We also have the following formulas: Also, in this paper, in order to simplify the proof of the theorems, we write O(z) as an expression that there exists a uniform constant C > 0 such that |O(z)| ≤ Cz.

Additive Eigenvalue Problem of the Laplace Operator with the Prescribed Contact Angle Boundary Value
In this section, we study the additive eigenvalue problem of the Poisson equation with the prescribed contact angle boundary value and prove eorem 1.
To prove eorem 1, one can find that we cannot get the C 0 estimate of the solutions to (1) since the solutions can differ from each other at least by a constant; thus, we cannot use the classical methods in partial differential equations to conclude the existence of the solutions. And we would like to settle the following perturbed equations and finally let the parameter ϵ go to zero: in Ω, where ] is the inner unit normal vector field along zΩ. For this equation with fixed ϵ, the gradient estimate is already derived in [19], and the C 0 estimate for some uniform constant M > 0 can also be deduced by following the same method in [25]. So for fixed ϵ, we can conclude the existence of the solution to (11) by Schauder theory, and we denote by u ϵ (x) the solution to (11).
In the following, we follow [20,22] to adopt a blow-up technique to give the uniform C 0 estimate of v ϵ .

Lemma 1.
Let Ω be a bounded domain in R n with a smooth boundary, n ≥ 2. If u ϵ (x) is the smooth solution to (11), v ϵ (x) is defined as above, and then there is a constant Proof. If this is not the case, without loss of generality, we assume that Let and w ϵ satisfies the following equation: where b ϵ � A − 1 ϵ for simplicity. To proceed with the proof, we need the following interior gradient estimate for the Poisson equation. e conclusion is known, and we omit the proof.

Lemma 2. Let Ω ⊂ R n be a bounded domain, n ≥ 2. If w satisfying |w| ≤ M for some positive constant M is the smooth solution to
we then have for any Ω ′ ⊂⊂ Ω e following lemma concludes the gradient estimate near the boundary.
Proof. Denote by v � where h and α are to be determined later, We may assume the maximum of Φ on Ω σ 0 occurs at some point y 0 for some σ 0 ≤ σ 1 .
Case 2 (y 0 ∈ zΩ). Choose a proper coordinate at y 0 ; let n denote the unit inner normal derivative and 1 ≤ i ≤ n − 1 denote the tangential derivative. en at y 0 , Hence, By a direct computation, we have where we denote by k ij the Weingarten matrix. Differentiating w n along zΩ, we obtain for i � 1, 2, . . . , n − 1 Furthermore, using we get Complexity Plugging (26) into (23), we have en, Without loss of generality, we may assume that v is large such that if α is chosen large enough determined by the geometry of zΩ and θ, the right-hand side of the above inequality will be positive which shows that this case will not occur at all. Case 3. (y 0 ∈ Ω σ 0 ). We take a special coordinate around y 0 such that w 1 � |Dw|, w l � 0(l � 2, 3, . . . , n), and w ij is diagonal for i, j ≥ 2. By the assumption, we have erefore, From the first-order condition (29), we get On the contrary, erefore, for i > 1, we get Denote without of loss of generality, then we may assume that |Dw| is large such that T 1 is larger than 0 due to the assumption that |cos θ| ≤ b < 1. erefore, And for i � 1, we get 4 Complexity A direct calculation shows that and thus by equation (19), For the term I 1 , For the term I 2 , and for the last term I 3 , Combining the results of I 1 , I 2 , and I 3 , we have en, we have Let h(w) � − ln cos(w/M), then erefore, we have at y 0 which shows that en by a standard discussion, we have And this finishes the proof of Lemma 3. Now we can proceed with the proof of Lemma 1.