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

Authors: Hongmei Li; Peihe Wang
Journal: Complexity
 (2020)

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## Abstract

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.


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## Body

## 1. 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 data-driven cloud service recommendation, etc. One can refer to the related references such as [1–15]. 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.The 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.
LetΩ be a bounded domain in ℝn with a smooth boundary, n≥2, then ν is the inner normal vector of ∂Ω. For any θ∈C∞Ω¯, and cosθ≤b<1, there exist a unique τ∈ℝ and a smooth function w∈C∞Ω¯ solving(1)Δu=τ,inΩ,∂u∂ν=−cosθ1+Du2,on∂Ω.Moreover, the solution to (1) is unique up to a constant.Our work is motivated by studies [20–22]. 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(2)χt+Fx,∇χ,∇2χ=λ,inΩ×0,∞,χx,0=χ0x,onΩ¯×0,Lx,∇χ=μ,on∂Ω×0,∞,where F and L are at least continuous functions defined, respectively, on Ω¯×ℝn×Sn and Ω¯×ℝn and got a convergence result χ⟶u∞, as t⟶+∞ uniformly in Ω¯, where u∞ is a solution to(3)Fx,Du,D2u=λ,inΩ×0,∞,Lx,Du=μ,on∂Ω×0,∞,under the assumption that F and L satisfy several structure conditions. Especially, we remark that the condition(4)Lx,p+sνx−Lx,p≥δs,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(5)ut−F∇2u=0,inΩ×0,∞,ux,0=u0x,onΩ¯×0,∂u∂ν=φx,on∂Ω×0,∞,and then in [23], Huang and Ye concluded the large time behavior of the solution in the classical senses. They 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:(6)F∇2u=λ,inΩ,∂u∂ν=φx,on∂Ω.A similar convergence result was obtained by Ma et al. [24] for the graphic mean curvature flow with the Neumann boundary condition:(7)ut=∑i,j=1nδij−uiuj1+∇u2uij,inΩ×0,∞,uν=φx,on∂Ω×0,∞,ux,0=u0x,onΩ¯,where Ω is a strictly convex bounded domain in ℝn with the smooth boundary for n≥2and u0x and φx are smooth functions satisfying u0,ν=φxon∂Ω. They proved that, up to a constant, the solutions converge to a translating solution λt+w. In other words, w,λ is a solution to(8)∑i,j=1nδij−uiuj1+∇u2uij=λ,inΩ,uν=φx,on∂Ω.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 Theorem1, we actually can follow the procedure in [24] to get the uniform gradient estimate without the C0 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.SupposeΩ is a bounded domain in ℝn, n≥2, ∂Ω∈C3. Set(9)dx=distx,∂Ω,Ωσ=x∈Ω:dx<σ.Then, there exists a positive constantσ1>0 such that ∀σ≤σ1,dx∈C3Ωσ¯. As mentioned by Lieberman [27] or Simon and Spruck [28], we can extend ν as Dd in Ωσ which is a C2 vector field. We also have the following formulas:(10)∇ν+∇2ν≤C˜n,Ω,inΩσ,ν=1,inΩσ.Also, in this paper, in order to simplify the proof of the theorems, we writeOz as an expression that there exists a uniform constant C>0 such that Oz≤Cz.


## 2. 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 Theorem1.To prove Theorem1, one can find that we cannot get the C0 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:(11)Δu=ϵu,inΩ,∂u∂ν=−cosθ1+Du2,on∂Ω,where ν is the inner unit normal vector field along ∂Ω. For this equation with fixed ϵ, the gradient estimate is already derived in [19], and the C0 estimate(12)ϵuϵ≤Mfor 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 uϵx the solution to (11).Letvϵx=uϵx−uϵx0 with some fixed x0∈Ω. In the following, we follow [20, 22] to adopt a blow-up technique to give the uniform C0 estimate of vϵ.Lemma 1.
LetΩ be a bounded domain in ℝn with a smooth boundary, n≥2. If uϵx is the smooth solution to (11), vϵx is defined as above, and then there is a constant A0>0, independent of ϵ, such that(13)vϵC0Ω≤A0.Proof.
If this is not the case, without loss of generality, we assume that(14)Aϵ=vϵC0Ω¯⟶+∞asϵ⟶0+.
Letwϵ=vϵ/Aϵ, then(15)wϵx0=0,wϵC0Ω¯=1,and wϵ satisfies the following equation:(16)Δw=ϵw+bϵuϵx0,wν=−bϵ2+Dw2cosθ,where bϵ=Aϵ−1 for simplicity.
To proceed with the proof, we need the following interior gradient estimate for the Poisson equation. The conclusion is known, and we omit the proof.Lemma 2.
LetΩ⊂ℝn be a bounded domain, n≥2. If w satisfying w≤M for some positive constant M is the smooth solution to(17)Δw=fx,w,inΩ,we then have for any Ω′⊂⊂Ω (18)supΩ′Dw≤CM,distΩ′,∂Ω,diamΩ,fC1Ω×−M,M.The following lemma concludes the gradient estimate near the boundary.Lemma 3.
LetΩ⊂ℝn be a bounded domain with a smooth boundary and ν be the inner unit normal vector field along ∂Ω, n≥2. If w satisfying w≤M for some positive constant M is the smooth solution to(19)Δw=ϵw+buϵx0,wν=−b2+Dw2cosθ,where 0<b≤1 is a constant, we then have for σ≤σ1 (20)supΩσDw≤CM,Ω,θC2Ω¯.Proof.
Denote byv=b2+Dw2 η=v+wνcosθ=v+∑i=1nwidicosθ. Let Φ=logη+hw+αd, where handα are to be determined later,
We may assume the maximum ofΦ on Ω¯σ0 occurs at some point y0 for some σ0≤σ1.Case 1.
(y0∈∂Ωσ0∩Ω). The estimate follows from the interior gradient estimate stated in Lemma 2.Case 2.
(y0∈∂Ω). Choose a proper coordinate at y0; let n denote the unit inner normal derivative and 1≤i≤n−1 denote the tangential derivative. Then at y0,(21)dn=1,di=0,i=1,…,n−1.
Hence,(22)0=Φi=ηiη+h′wi+αdi=ηiη+h′wi,i=1,2,…,n−1,0≥Φn=ηnη+h′wn+αdn=ηnη+h′wn+α.
By a direct computation, we have(23)ηn=vn+wnncosθ+wncosθn=∑i=1nwiwinv+wnncosθ+wncosθn=∑i=1n−1wiwinv+wnwnnv+wnncosθ+wncosθn=∑i=1n−1wiwniv+∑i,j=1n−1wikijwjv+wncosθn,where we denote by kij the Weingarten matrix.
Differentiatingwn along ∂Ω, we obtain for i=1,2,…,n−1 (24)wni=−vcosθi=−vicosθ−vcosθi=−ηi−wnicosθ−wncosθicosθ−vcosθi=−ηicosθ+wnicos2θ+wncosθcosθi−vcosθi.
Furthermore, using(25)0=Φi=ηiη+h′wi,we get(26)wni=ηh′wicosθ−v1+cos2θcosθisin2θ.
Plugging (26) into (23), we have(27)ηn=∑i=1n−1wiwniv+∑i,j=1n−1wikijwjv+wncosθn=∑i=1n−1wiηh′wicosθ−v1+cos2θcosθivsin2θ+∑i,j=1n−1wikijwjv+wncosθn.
Then,(28)0≥Φn=∑i=1n−1wiηh′wicosθ−v1+cos2θcosθiηvsin2θ+∑i,j=1n−1wikijwjηv+1ηwncosθn+h′wn+α=h′cosθ∑i=1n−1wi2vsin2θ−1+cos2θ∑i=1n−1wicosθiηsin2θ+∑i,j=1n−1wikijwjηv+1ηwncosθn+h′wn+α=h′cosθv2sin2θ−b2vsin2θ−1+cos2θ∑i=1n−1wicosθiηsin2θ+∑i,j=1n−1wikijwjηv+1ηwncosθn+h′wn+α=h′vcosθ−h′b2cosθvsin2θ−1+cos2θ∑i=1n−1wicosθiηsin2θ+∑i,j=1n−1wikijwjηv+1ηwncosθn+h′wn+α=−h′b2cosθvsin2θ−1+cos2θ∑i=1n−1wicosθiηsin2θ+∑i,j=1n−1wikijwjηv+1ηwncosθn+α.
Without loss of generality, we may assume thatv is large such that if α is chosen large enough determined by the geometry of ∂Ω 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.
(y0∈Ωσ0). We take a special coordinate around y0 such that w1=Dw,wl=0l=2,3,…,n,andwij is diagonal for i,j≥2. By the assumption, we have(29)0=Φi=ηiη+h′wi+αdi,(30)0≥Φij=ηijη−ηiηjη2+h′wij+h″wiwj+αdij.
Therefore,(31)0≥ΔΦ=Δηη−∑i=1nh′wi+αdi2+αΔd+h′Δw+h″Dw2=Δηη+h′Δw+h″−h′2Dw2−2h′αd1w1+αΔd−α2Dd2=I¯+II+III+IV+V.
From the first-order condition (29), we get(32)ηi=−ηh′wi+αdi.
On the contrary,(33)ηi=vi+wνcosθi=vi+∑l=1nwldlcosθi=∑l=1nwlwliv+∑l=1nwlidlcosθ+∑l=1nwldlcosθi=∑l=1nwlv+dlcosθwli+∑l=1nwldlcosθi.
Therefore, fori>1, we get(34)dicosθwii+w1v+d1cosθw1i+w1d1cosθi=−αηdi.
Denote(35)T1=w1v+d1cosθ,without of loss of generality, then we may assume that Dw is large such that T1 is larger than 0 due to the assumption that cosθ≤b<1. Therefore,(36)w1i=−dicosθwiiT1−ηαdiT1−d1cosθiw1T1=−dicosθwiiT1+Ov.
And fori=1, we get(37)w11=−∑l=2ndlcosθw1lT1−ηh′w1+αd1T1−d1cosθ1w1T1=∑l=2ndlcosθdlcosθwll/T1+OvT1−ηh′w1+αd1T1−d1cosθ1w1T1=∑l=2ndlcosθ2T12wll+Ov2.
A direct calculation shows that(38)ηij=∑l=1nwlv+dlcosθwlij+∑l=1nwlv+dlcosθjwli+∑l=1ndlcosθiwlj+∑l=1ndlcosθijwl=∑l=1nwlv+dlcosθwlij+∑l=1nwljv−wlvjv2+dlcosθjwli+∑l=1ndlcosθiwlj+∑l=1ndlcosθijwl,and thus by equation (19),(39)Δη=∑l,i=1nwlv+dlcosθΔwl+∑l,i=1nwli2v−∑i=1nw12w1i2v3+2∑l,i=1ndlcosθiwli+w1Δd1cosθ≥∑l,i=1nwli2v−∑i=1nw12w1i2v3+2∑l,i=1ndlcosθiwli+w1Δd1cosθ+w1d1cosθ=I1+I2+I3.
For the termI1,(40)I1=∑l,i=1nwli2v−∑i=1nw12w1i2v3=b2w112v3+1v+b2v3∑i=2nw1i2+∑i=2nwii2v≥∑i=2nwii2v.
For the termI2,(41)I2=2d1cosθ1w11+2∑l=2nd1cosθl+dlcosθ1w1l+2∑l=2ndlcosθlwll=2d1cosθ1∑l=2ndlcosθ2T12wll+Ov2+2∑l=2ndlcosθlwll−2∑l=2nd1cosθl+dlcosθ1dlcosθwllT1+Ov=∑l=2nO1wll+Ov2,and for the last term I3,(42)I3=Ov.
Combining the results ofI1, I2, and I3, we have(43)I=I1+I2+I3≥∑l=2nwll2v+O1wll+Ov2≥Ov2.
Then, we have(44)I¯=Δηη≥Ov.
Lethw=−lncosw/M, then(45)II=h′Δw=O1,III=h″−h′2Dw2=1M2Dw2,IV=−2h′αd1w1=2h′αd1Dw≤12M2Dw2+O1,V=αΔd−α2Dd2=O1.
Therefore, we have aty0(46)0≥ΔΦ≥Ov+12M2Dw2+O1,which shows that(47)Dw2y0≤C.
Then by a standard discussion, we have(48)Dw2x≤C,∀x∈Ω¯σ0.
And this finishes the proof of Lemma3.
Now we can proceed with the proof of Lemma1.Proof of Lemma 1.
Based on the estimates in Lemma2 and Lemma 3, we can use Schauder theory to consider the limit behavior of problem (16) and finally get that(49)Δw=0,inΩ,wν=−Dwcosθ,on∂Ω.
It shows that the maximum ofw only occurs on the boundary, denoted by x¯. Then at this point, the tangential part of Dw is zero and wν≠0. It follows that(50)cosθ=1.
But this contradicts the conditioncosθ≤b<1, and thus, the proof of Lemma 1 is finished.Proof of Theorem 1.
Sincevϵx=uϵx−uϵx0, the equation vϵ satisfies(51)Δvϵ=ϵvϵ+uϵx0,vνϵ=−1+Dvϵ2cosθ.
To consider this equation, because of (12), we can apply Lemma 2 and Lemma 3 to conclude the uniform gradient estimate of vϵ. Combined with Lemma 1, we immediately derive the uniform C1 bound of vϵ independent of ϵ. And then follows the uniform Ck,α bound of vϵ, for k∈Z+,α∈0,1. Now we come to think about the process as ϵ goes to zero. Obviously, we can assume that vϵ will converge to a smooth function v in the Ck,α sense for k∈Z+,α∈0,1.
For the functionϵvϵ+uϵx0=ϵuϵx with the parameter ϵ, we know by (12) that they are uniformly bounded with their gradient; thus by the Arzela–Ascoli theorem, there is a subsequence, denoted also by ϵuϵx, converging to a C1 function v0x satisfying Dv0x=limϵ⟶0ϵDvϵ+uϵx0=0, where we used the fact that Dvϵ is uniformly bounded. Thus, v0x is a constant function τ. That is to say, we have solved the following additive eigenvalue problem:(52)Δv=τ,inΩ,∂v∂ν=−cosθ1+Dv2,on∂Ω.
For the uniqueness, we can assume thatv¯,τ¯ is also a solution to (1), we may suppose that τ≤τ¯, and then we have(53)Δv−v¯=τ−τ¯≤0,inΩ,∂v−v¯∂ν=−b⟶,Dv−v¯cosθ,on∂Ω,where b⟶=∫01Dtv+1−tv¯/1+Dtv+1−tv¯2dt is a vector with b⟶<1. We then can follow the discussion of (49) to discuss the minimum point of v−v¯ and finally deduce that v−v¯ must be a constant and τ=τ¯. This finishes the proof of Theorem 1.Remark.
From Introduction, we can know that the additive eigenvalue problem is usually related to the large time behavior of the corresponding parabolic equation with the same kind of boundary value. Actually, we will come to consider this problem in the forthcoming paper.



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