Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part II
Xuan Hien Nguyen

TL;DR
This paper proves the existence of certain self-similar surfaces under mean curvature flow, advancing the construction of new complete embedded solutions with specific boundary conditions and symmetries.
Contribution
It establishes the existence of solutions to the Dirichlet problem for self-similar surfaces over punctured planes, contributing to the construction of novel embedded self-similar surfaces.
Findings
Existence of solutions under small boundary conditions
Solutions satisfy prescribed symmetries
Progress towards new embedded self-similar surfaces
Abstract
We study the Dirichlet problem associated to the equation for self-similar surfaces for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries. This is the second step towards the construction of new examples of complete embedded self similar surfaces under mean curvature flow.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Advanced Mathematical Modeling in Engineering · advanced mathematical theories
Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part II.
Xuan Hien Nguyen
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, 45221
Abstract.
We study the Dirichlet problem associated to the equation for self-similar surfaces for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries. This is the second step towards the construction of new examples of complete embedded self similar surfaces under mean curvature flow.
Key words and phrases:
mean curvature flow, self-similar, singularities
2000 Mathematics Subject Classification:
Primary 53C44
1. Introduction
This paper is the second one of a series of three articles describing the construction of new examples of complete embedded self-similar surfaces under mean curvature flow [8][7]. Our general strategy is inspired by Kapouleas’ article [4] and is described in detail in the previous installment. Let us recall it briefly: to construct a new self-similar surface, we take two known examples suitably positioned and replace a neighborhood of their intersection with an appropriately bent scaled Scherk’s singly periodic surface. The procedure is called desingularization.
The resulting surface is not smooth; however it is a good approximate solution. The next task is to find a small function whose graph over it satisfies the self-similar surface equation
[TABLE]
where is the mean curvature and is the normal vector so that the mean curvature vector is . The sign of is chosen so that the mean curvature of a convex surface is positive. Before considering graphs of functions on the entire surface, we have to work locally and study the Dirichlet problems with small boundary conditions corresponding to (1) for graphs of functions on the different pieces.
In the first part [8], we found that a perturbation of an appropriately scaled and bent Scherk surface satisfies (1) and can be used to desingularize the intersection of a cylinder and a plane or a sphere and a plane. In this article, we study the Dirichlet problem corresponding to (1) for functions on the outer plane, which is the Euclidean plane with a disk removed. We show that the Dirichlet problem possesses a solution if the boundary conditions on the circle are small enough. Our methods here are different from the one used in Part I. Since the equation (1) can be written with a single set of coordinates on the outer plane, our tools come mainly from partial differential equation theory, with some interior estimates established by Ecker and Huisken [1] for the Mean Curvature Flow. The third article is to discuss the gluing of the solutions on the different pieces in a manner to obtain a smooth complete embedded self-similar surface.
1.1. Main result
Let and , consider a function satisfying the symmetries
[TABLE]
Theorem 1**.**
There is an depending on and such that for any with and satisfying the symmetries above, there is a function on such that
[TABLE]
In addition, We can choose the constant uniformly for all such that .
Here, on means that in polar coordinates. This is a slight abuse of notation which does not induce confusion. Note that the conditions (5) imply that is odd with respect to rotations of degrees with respect to the -axis and even with respect to reflections across the planes , which are exactly the symmetries imposed in Part I [8].
Computing the mean curvature and the normal vector of the graph of using coordinates in , we find that (3) is equivalent to
[TABLE]
where . This is our main equation. Note that our domain is unbounded and that the coefficient in front of is positive, so the standard maximum principle is not applicable directly.
We first study the properties of the linear operator associated to . From solutions to the linear problem, we then construct a sub and supersolution to the quasilinear equation .
The solution to the elliptic equation (6) is found as a limit for time going to infinity of a solution to the parabolic equation . For initial conditions with bounded gradient, standard theory on parabolic equations assures the existence of a solution to the initial value problem on a short time interval . The constructed sub and supersolution serve as barriers for solutions of the parabolic equation and allow us to control the growth of at infinity in the space variable for all time. Exploiting the close relation between the parabolic equation and the mean curvature flow, we use interior estimates from the latter to bound derivatives of our solution to the former away from the boundary. Loosely speaking, this gives us control of the derivatives outside of a bounded annulus around . To bridge the gap, we invoke standard parabolic theory on bounded sets and obtain uniform estimates for derivatives of for all . This implies that the solution has to exists for all time . We conclude that a subsequence of tends to a solution of as goes to infinity by proving a monotonicity formula.
Some sections of this paper appeared in the author’s thesis, written under the direction of Sigurd Angenent at the University of Wisconsin-Madison. The author is indebted to him for invaluable discussion.
2. Definitions
2.1. Mean curvature flow
The properties of solutions to (3) are closely related to properties of solutions to the mean curvature flow (MCF), which will be used intensively. Let us therefore define it.
Let be a one parameter family of immersions of -dimensional smooth hypersurfaces in . We say that is a solution to the mean curvature flow if
[TABLE]
is satisfied for some initial data . Here is the mean curvature vector of the hypersurface at .
If the hypersurfaces can be written locally as graphs of a function over a domain in the - plane, the quasilinear equation
[TABLE]
is equivalent to (7) up to tangential diffeomorphisms.
2.2. Weighted Hilbert Sobolev spaces , and
We consider the Hilbert space
[TABLE]
where is an open subset of (not necessarily bounded) and is the Gaussian measure We write and for the norm and inner product of . We also define the Hilbert space
[TABLE]
The inner product of is Let be the closure of in . Even when is unbounded the inclusion is compact.
3. The Linear Operator
Let us define sections of the outer plane by
[TABLE]
and their corresponding Hilbert spaces
[TABLE]
In this section, we study the properties of , the linear operator associated to defined in (6),
[TABLE]
3.1. Eigenvalues and eigenfunctions
Consider the operator
[TABLE]
on . The compactness of the inclusion yields the following theorem:
Theorem 2**.**
The operators and have a countable set of eigenvalues having no limit point except possibly and a corresponding -orthogonal basis of eigenfunctions .
Proof.
Theory on compact bounded self-adjoint operators in Hilbert spaces gives us the result for . The result for then follows immediately. ∎
Consider eigenfunctions of the form satisfying the symmetries
[TABLE]
There is an eigenvalue such that
[TABLE]
Separation of variables and the symmetry conditions (10) on imply that the only possibilities for are with , . Therefore
[TABLE]
We have
[TABLE]
is a linear map on the vector space and it is upper diagonal in the basis . The operator has a zero eigenvalue if one of the entry on the diagonal vanishes, i.e. if . We therefore found some of the eigenvalues of our original operator ,
[TABLE]
where is the set of natural numbers. For each value of the corresponding eigenfunction is of the form
[TABLE]
where is a polynomial of degree .
We now show that the eigenvalues values given by (11) are in fact all the possible eigenvalues of . This is done by proving the following theorem:
Theorem 3**.**
The set of eigenfunctions (12) corresponding to the eigenvalues (11) forms a basis for the Hilbert space .
Before we start the proof, note that for each fixed and ,
[TABLE]
since they are eigenfunctions corresponding to different eigenvalues. Hence, By the change of variable ,
[TABLE]
The polynomials , which are closely related to the general Laguerre polynomials, are therefore orthogonal in , the -space weighted by . In addition, they span the functions , . The fact that is a maximal orthogonal set in follows from the theorem p. 333 in [9] which is recalled below.
Theorem 4** (Theorem p. 333 [9]).**
The orthonormal sequence of polynomials attached to a mass distribution function on the interval , is complete in whenever there exists a number such that the integral
[TABLE]
exists.
Proof of Theorem 3.
It suffices to show that the set of all finite linear combinations of members of is dense. Let . Since is periodic in and for almost every , we can expand in Fourier series for almost every . The symmetries of imply . Using the change of variable , we write
[TABLE]
The -norm of is finite, therefore
[TABLE]
Hence, the functions ’s are in for . For every , there is a so that
[TABLE]
We can approximate by a linear combination of , since is a maximal orthogonal set . More precisely, for every and every , there is a linear combination of ’s denoted by so that
[TABLE]
Define to be the linear combination of elements of given by . Then
[TABLE]
This shows that every function in can be approximated arbitrarily closely by a linear combination of functions of . ∎
3.2. Poincaré inequality
As an immediate consequence of Theorem 3, we have the following inequality for :
[TABLE]
where is the lowest eigenvalue of the operator . This implies a Poincaré inequality
[TABLE]
for every domain .
3.3. Dirichlet problem
Let be defined as in (9).
Lemma 5**.**
For any , the equation
[TABLE]
possesses a weak solution .
Proof.
It is a classical variational argument for the functional J(u)=\int_{\Omega_{R,N}}\Bigl{\{}\frac{1}{2}|Du(\xi)|^{2}-\frac{1}{2}u(\xi)^{2}+g(\xi)u(\xi)\Bigr{\}}\;dm(\xi), over and using the Poincaré inequality (15). ∎
3.4. Maximum Principle.
Let be any domain that is a (not necessarily proper) subset of .
Definition 6**.**
We say that satisfies on if its positive part .
Theorem 7** (Maximum Principle).**
Let . Suppose satisfies
[TABLE]
then in .
Proof.
First note that is a solution of . By the hypotheses, we have
[TABLE]
Define the function such that Since and are uniformly bounded, . Moreover, satisfies
[TABLE]
Since and , we have Hence,
[TABLE]
Take in (17). Note that we can do this since and , so . Therefore
[TABLE]
since on the support of . Using and , we get
[TABLE]
and
[TABLE]
by the Cauchy Schwarz inequality. Our Poincaré inequality (15) implies that
[TABLE]
A simple computation shows that for , . Therefore
[TABLE]
But , so ; this means that and in . ∎
3.5. Boundary value problem
Let with the symmetries (2): . We will use the notation throughout Section 3.
Definition 8**.**
We say that is a solution to the problem
[TABLE]
if satisfies
[TABLE]
where is a function in ( resp.) with and for , ( resp.).
It follows from Lemma 5 and the Maximum Principle (Theorem 7) that the solution to in , on exists and is well defined, in other words it is unique and independent of the choice of .
Denote by the extension of to given by
[TABLE]
The function is a solution to boundary value problem
[TABLE]
Moreover, it is smooth by standard elliptic theory.
3.6. Behavior of solution at infinity
Let be the solution to the problem in , on . We can choose in Definition 8 satisfying with in Let and consider the function
[TABLE]
Since , therefore on . It follows from the Maximum Principle that in , hence An similar argument with gives a lower bound on , so
[TABLE]
Therefore, we also have
[TABLE]
3.7. Bounds on the first and second derivatives of the solution
Let be the solution to (18) from section 3.5.
Lemma 9**.**
There is a constant independent of and so that
[TABLE]
Proof.
Step 1: Estimate away from the boundary.
The function is smooth in by elliptic theory. A computation shows that the function
[TABLE]
satisfies the heat equation The theory for parabolic equations gives us the following estimates on the derivatives of ,
[TABLE]
where is a constant independent of or and as long as exists in . This is true provided
[TABLE]
In particular, it is true if and . From (22), we get
[TABLE]
The last factor of the right hand side can be estimated using equation (20) to obtain
[TABLE]
Using the notation , we have
[TABLE]
Similarly, we get an estimate on the second derivative
[TABLE]
Therefore
[TABLE]
Since , the estimate (23) is valid for in particular.
Step 2: Estimates up to the boundary.
Consider the equation The domain is bounded so global regularity results for elliptic equations from [3] imply that
[TABLE]
where is a function in such that .
Note that we can bound the first four derivatives of for by an argument similar to the one in step 1,
[TABLE]
It is then clear that we can find a constant independent of and , and a function for which
[TABLE]
With this choice of , equation (24) gives us
[TABLE]
Using Sobolev’s inequality, (25), (26) and (20), we get
[TABLE]
Combining (23) and (27), we obtain the desired result. ∎
4. Finding sub and supersolutions
This section is devoted to the construction of a subsolution and a supersolution to the problem (6)-(4), or equivalently, to
[TABLE]
with small . Let us discuss the strategy for finding a supersolution first. Since the boundary data is of order , we write ,
[TABLE]
and decompose into the two terms , where
[TABLE]
We discussed the existence of such a function in section 3.6. Choosing an appropriate function is done below.
4.1. Preliminary computations for finding a supersolution
In order to get a supersolution, we want to satisfy Roughly estimating by and bounding the derivatives of using Lemma 9, we are looking for a so that
[TABLE]
where . Define A simple computation gives
[TABLE]
This makes a perfect candidate for .
Remark 10*.*
The term has to counterbalance the contribution from the nonlinear term. Therefore, we need to be negative. This justifies the imposed lower bound .
4.2. Existence of a supersolution and a subsolution
Let be a small constant. Consider the function
[TABLE]
where is a constant to be chosen later.
[TABLE]
We have and , which combined with (21), give us
[TABLE]
Here we used the notation . We replace by , with a constant to be chosen later, to get
[TABLE]
If is small enough so that the equation (32) becomes
[TABLE]
Hence, to have a supersolution, it suffices to find and that satisfy
[TABLE]
If we take so that , both inequalities are true for and . Recall that , therefore, for and so that
[TABLE]
the function is a supersolution with boundary condition . A similar argument shows that is a subsolution with boundary condition . If we denote by and , we just proved the following result.
Lemma 11**.**
Assume and denote by , the plane with a hole of radius at the origin. Let be a function in with the symmetries (2), a solution to the linear equation , and a constant (independent of ) so that
[TABLE]
If satisfies
[TABLE]
then the function , where , satisfies
[TABLE]
and the function satisfies
[TABLE]
From the bounds on the derivatives of and the definition of and , we have
Corollary 12**.**
Let and be defined as in the lemma above. Then there exists a constant depending only on and so that
[TABLE]
5. The Parabolic Equation
In this section, we prove that there exists a solution to the parabolic equation for all time for well chosen initial and boundary conditions and respectively.
To avoid confusion, let us fix notations: we denote by the ordinary derivative with respect to -th coordinate or of a spatial variable and by the gradient of the function with respect to spatial variables. The notation or is reserved for the derivative with respect to time.
5.1. Initial condition and short time existence
For the initial condition to the parabolic equation, we choose a function in that stays between the sub and supersolutions constructed in Lemma 11 and has bounded derivatives
[TABLE]
Moreover, we assume that satisfies the following symmetries and compatibility conditions
[TABLE]
Let us define the domains and . We consider the following corresponding problem for the mean curvature flow
[TABLE]
for some initial condition . Standard parabolic theory assures the existence of a smooth short time solution to the boundary value problem (35) with initial condition . Moreover, if we denote the maximal time interval in which exists by , we have
[TABLE]
For more details, we refer to [2].
In order to show that the solution exists for all time , we argue by contradiction and assume that then show that is uniformly bounded for all . Standard parabolic theory implies that the solution can then be continuated past , which contradicts the maximality of . The rest of this section is devoted to the proof of a uniform bound on .
The function defined by
[TABLE]
satisfies
[TABLE]
where we used the change of variables and .
Lemma 13**.**
Let be a smooth solution of (38a) in with boundary condition (38b), then
[TABLE]
Proof.
To each solution to (38) corresponds a solution to the problem (35) via the formula (37). This lemma is therefore a consequence of (36). ∎
5.2. Bounds on the function
Here, we show that a solution to the problem (38) stays between the subsolution and the supersolution :
Lemma 14**.**
Let be a solution to the problem (38), then
[TABLE]
Proof.
The proof is a slight modification of the proof of a Maximum Principle for parabolic equations. Let be as in Corollary 12. For small , consider the function , where and is to be chosen later. Note that becomes negative as tends to infinity. We have and D_{ij}\psi=\Bigl{[}-\frac{\varepsilon}{|\xi|}(\delta_{ij}-\frac{\xi_{i}\xi_{j}}{|\xi|^{2}})+\varepsilon^{2}\frac{\xi_{i}\xi_{j}}{|\xi|^{2}}\Bigr{]}\psi. A computation shows that satisfies
[TABLE]
Fix . The estimates in Corollary 12 and Lemma 13 guarantee the existence of a large constant such that for , . Denote by the annulus . On the boundary of the cylinder , we have
[TABLE]
Suppose that for some point in the cylinder . There is a first time for which there is a point with . At this point , is positive definite, and . At , the equation (41) yields
[TABLE]
The last term can be estimated by To bound the penultimate term, note that
[TABLE]
therefore, \left\lvert\psi\bigl{(}g^{ij}(Du_{-})-g^{ij}(Du)\bigr{)}D_{ij}u_{-}\right\rvert\leq 12\sqrt{2}\varepsilon^{2}K_{2} at by the Mean Value Theorem. We obtain
[TABLE]
The choice of makes the last expression negative. This is not possible. We have therefore shown that for . The argument stays valid for any , thus
[TABLE]
Letting tend to [math], we obtain for Since is arbitrary, we have
[TABLE]
The inequality is proved in a similar way. ∎
5.3. Bound on the first derivative
We prove the bound in two steps. First, we use Lemma 14 and interior estimates from the mean curvature flow to bound . The interior estimate on gives us estimates for for points staying farther away as time progresses. The second step is to consider an annulus , with large enough to have bounds for on from the first step. A maximum principle for the parabolic equation satisfied by yields estimates in the interior of the annulus, and therefore on the whole domain.
Lemma 15**.**
Let be a smooth solution to (38a) in the cylinder with boundary conditions (38b). There is a constant depending only on the boundary conditions and such that
[TABLE]
for all times for which exists, and
Proof.
*Step 1. Away from the boundary. *
Let be the corresponding solution to the MCF defined by equation (37). We use the interior estimate below established by Ecker and Huisken:
Theorem 16** (Theorem 2.3, p551 [1]).**
The gradient of the height function satisfies the estimate
[TABLE]
where is a ball in and is the dimension of the graph of .
The first factor is estimated by
[TABLE]
To bound the second factor, we recall that therefore Corollary 12 gives us
[TABLE]
The function is defined on the domain , in particular, it is defined for at any time . Let and . The estimates (44) and (45) yield,
[TABLE]
Note that Du\bigl{(}\frac{x}{\sqrt{2(1-t)}},-\frac{1}{2}\ln(1-t)\bigr{)}=Dv(x,t). With the change of variables and , we get
[TABLE]
Withouth loss of generality, we can assume that .
Step 2: Up to boundary, using a Maximum Principle.
A computation shows that the function satisfies the parabolic equation
[TABLE]
where and . Denote by , then
[TABLE]
Choose . It follows from Lemma 13 that the coefficients in (47) are bounded on . Moreover, from (46), we know that is bounded in the cylinder . A Maximum Principle for parabolic equations on unbounded domains (see Theorem 8.1.4 in [5] for example) implies that the function in is smaller than its supremum on the parabolic boundary . Since is arbitrary, the result is valid on the time interval as well,
[TABLE]
5.4. Bound on and away from boundary
Lemma 17**.**
Let be a solution of the problem (38) in , then we have the following bounds for the second and third derivatives of :
[TABLE]
[TABLE]
where and denotes different constants depending only on , and .
Remark 18*.*
The inequality (48) does depend on through the constant ; however the dependence is very loose. In the proof, we will see that can be chosen to be for any time . Therefore, the estimate is uniform in time once we know that the solution exists past some small time .
Remark 19*.*
and depend on on the right hand side, but we are primarily interested in finding a bound for approaching so there is no need for a better estimate near initial time.
Proof.
Let us tackle the second derivative first. The estimate for is immediate. For , we use the change of variables and look at the function
[TABLE]
which satisfies the mean curvature flow (35a). We are interested in time such that . We can use the following interior estimate on the curvature from Ecker-Huisken [1] as long as exists in the domains mentioned. Define to be .
Lemma 20** (Corollary 3.2 [1]).**
Let and and . For , we have the estimate
[TABLE]
where denotes the ball of radius centered at in the plane.
First, we show that exists in for , and : if , then and the corresponding satisfies Hence, for and all ,
[TABLE]
Bounds on the second fundamental form and on the first derivative yield a bound on the second derivative of :
[TABLE]
where denotes a different constant, also dependent on . Therefore
[TABLE]
In order to prove an estimate independent of time and outside a fixed annulus, note that for all , the function is also a solution of . Moreover, for so we have an estimate similar to (50) in this case also
[TABLE]
Let and , the inequality above yields
[TABLE]
for all , i.e. for all .
The bound on the third derivative is proved in a similar fashion using
[TABLE]
with the constant depending on and from Theorem 3.4 in [1]. ∎
5.5. Hölder continuity of near boundary
Let the time and denote by
[TABLE]
For a domain , denote by the set of all continuous functions in having continuous derivatives , in and by the set of functions on that are -Hölder continuous in and - Hölder continuous in .
We use the following theorem by Ladyženskaja et al.
Theorem 21** (Theorem 2.3, p533 [6]).**
Let be a solution of equation
[TABLE]
belonging to . Suppose that equation (51) is parabolic with respect to , i.e the inequality
[TABLE]
is fulfilled and the functions and are continuous and continuously differentiable with respect to all of their arguments in the domain
[TABLE]
Suppose also that there is a constant so that
[TABLE]
Then for some , the quantity is estimated by a constant depending only on , the quantities and , the norm and the norm in of the functions defining the boundary of . These same quantities also determine the exponent .
The coefficients of the parabolic equation
[TABLE]
in satisfy the hypotheses of Theorem 21. We therefore have a bound on that depends on , , and . From Lemma 17, we can estimate this last quantity uniformly in time, therefore,
[TABLE]
where only depends on time through .
5.6. Hölder continuity of near boundary
The coefficients of (38a) are in hence we can apply standard theory for linear parabolic equations with Hölder coefficients (Theorem 5.2 p320 [6] for example) to obtain
[TABLE]
The Remark 18 explains the loose dependence of our bounds on and . Let us emphasize it again: if the solution is known to exist past a small time , we can choose and the constants in all the estimates following (48) do not depend on .
5.7. Continuation of the solution
Combining (48) and (53), we have
[TABLE]
Moreover, Lemma 15 gives us on . These estimates can be transformed into uniform bounds on the first two derivatives of the corresponding solution to the mean curvature flow (35a) for time . Therefore the solution has a continuation existing for time with . This contradicts the maximality of , and proves that the solution to (38) exists for all time . Since none of the bounds on , or depended on , we have
[TABLE]
6. A Solution to the Elliptic Equation =0
Here, we show that there is a subsequence of that tends to a solution to the elliptic equation as .
Let us denote by the graph of over , by the position vector of a point on . From the equation , the variation of with respect to time is
[TABLE]
where is the unit vector in . It will be useful for what follows to decompose and into their tangent and normal components with respect to :
[TABLE]
where .
Consider the functional where is the Hausdorff measure on . We have
[TABLE]
Writing in terms of and as an integral over , we obtain
[TABLE]
By definition, for all time , so
[TABLE]
and
[TABLE]
Define for . For any constant , the ’s are uniformly bounded in . We can therefore extract a subsequence that converges in , . A Cantor diagonal argument gives us a further subsequence and a function such that as in the norm for every . Since every satisfies in and on the limit function inherits the same properties. On the one hand, equation (57) and the uniform bounds (54) and (55) tell us that pointwise. On the other hand, we know that pointwise. Hence, and satisfies in and , . We have therefore proved
Lemma 22**.**
Let and . Define to be the plane with a hole of radius . There is a depending on and such that for any with and satisfying the symmetries , there exists a smooth solution to the Dirichlet problem
[TABLE]
In addition, We can choose the constant uniformly for all such that .
7. Uniqueness
For the sake of clarity let us assume in this section. The same reasoning also works in the more general case although the estimates are more involved.
Let be our boundary condition, and suppose that , where is given by Lemma 22.
Theorem 23**.**
There is an so that, if and if and are two solutions to
[TABLE]
with then .
Before we start the proof, we need some a priori estimates on the solutions and .
7.1. A priori estimates
Because the solutions and above are not dependent on time, they satisfy the parabolic equation as well. Estimates similar to the ones from Lemmas 15 and 17 therefore hold in this case also, with some small modifications. If is a solution to the elliptic equation (58a), then the function
[TABLE]
is a solution to the mean curvature flow on . The estimate
[TABLE]
where is a ball in and is the dimension of the graph of , is valid. The last factor of the right hand side can be bounded as in the proof of Lemma 15; however, we do not have an initial condition independent of in this case, so right now, our estimates depend on the value of over a bounded set. Taking and , we obtain
[TABLE]
hence, by the change of variable ,
[TABLE]
An argument similar to the one in the proof of Lemma 17 gives us bounds on higher derivatives of away from the boundary
[TABLE]
with the constant depending on . Standard elliptic theory on quasilinear equations (see Theorem 13.4 or 13.7 [3] for example) assures the existence of constants and depending only on , and such that
[TABLE]
where is the annulus . The Hölder norm of the coefficient , seen as a function of , is uniformly bounded over the whole domain . We have a bound on by elliptic theory; therefore is also bounded.
7.1.1. The gradient achieves its maximum on the boundary .
Equation (47) from section 5 implies that satisfies
[TABLE]
where and . We will derive the following theorem as an application of the maximum principle:
Theorem 24**.**
Assume is a solution of (58), then achieves its maximum on the circle .
From the discussion above, is uniformly bounded and is a uniformly elliptic operator on . Moreover, the coefficients of are Hölder continuous with bounded Hölder norm on any compact set of .
Lemma 25**.**
If there is a function for which and as .
Proof.
We consider positive increasing functions . The operator is estimated by
[TABLE]
We have uniform bounds on and , therefore
[TABLE]
The solution to the differential equation is given by
[TABLE]
For the choice of boundary conditions , the function and its derivative are positive for . The last thing to verify is that tends to infinity as approaches infinity. This is true since for large , the dominant term in the expression for is of order . ∎
Proof of Theorem 24.
Let us consider the function , with as in Lemma 25 and . This function is negative for large values of . From equation (60), we have
[TABLE]
for any positive constant . By the maximum principle, does not achieve an interior maximum. Since the function goes to as goes to infinity, it has to achieve its maximum on the boundary ,
[TABLE]
Letting tend to [math], we obtain for . ∎
Our subsolution and supersolution provide barriers for the function , therefore
[TABLE]
with the constant independent of and . With this bound, we can run through the arguments in the beginning of this section again and obtain estimates for the second derivative that are independent of the solution .
7.2. Proof of uniqueness
Proof of Theorem 23.
Let and be two solutions as described in the hypotheses. We substract equation (58a) for from the equation for and use the notation to obtain
[TABLE]
The Mean Value Theorem implies that
[TABLE]
for some and where for . Denoting by the vector with components , , we have
[TABLE]
A computation shows that . From Section 7.1, we know that there exists a constant so that Moreover, the maximum of the first derivative of any solution to (58a) is achieved on the boundary hence and are controlled by and go to zero as tends to [math]. The same fact is true for by the definition of .
The maximum principle does not apply immediately to (61) because of the positive coefficient in front of . To circumvent this, we use a positive supersolution to (61) which grows faster than linearly at infinity. The existence of such a supersolution is given below and is the main difficulty in this proof.
Assuming such a positive supersolution exists, we define and look at
[TABLE]
The coefficient in front of in the explicit expression of is now , which is non positive, so the maximum principle applies to . It implies that for any subdomain of ,
[TABLE]
From the hypotheses, on and from the growth of at infinity, tends to [math] as tends to infinity. Choosing the domains to be increasingly large annuli, we obtain that . Since and , , therefore in the entire domain . Switching the roles of and we show that and conclude that in .
We are left to prove the existence of the function .
Claim 26**.**
For and small enough, there exists a function on satisfying
[TABLE]
The supersolution is found among functions depending on the radius only, , that are increasing . The operator applied to a radial function is estimated by
[TABLE]
Let us consider functions of the form
[TABLE]
with constants to be chosen later and . This function is positive on the set provided . To simplify the notations, we work with the case and imposed ; however, the following argument is also valid in general for with slightly more subtle estimates. The first and second derivatives are given by and . We then have
[TABLE]
By choosing small enough, we can assume that . A computation gives
[TABLE]
The coefficient of is negative if and . Such a choice of also guarantees that the coefficient of is negative. Therefore
[TABLE]
since . The right hand side of (62) is negative if we choose small enough and so that
[TABLE]
8. Remarks
Let and let be the unique solution to (58) such that .
Remark 27*.*
Because of uniqueness, the function has to satisfy the symmetries
[TABLE]
Taking , Theorem 1 from the introduction is then a corollary of Lemma 22 and Theorem 23.
Remark 28*.*
From the explicit formulas in Lemma 11, we have
[TABLE]
where the constant and the function do not depend on . Recall that is the constructed solution to the linear problem , . This implies that on the boundary circle , in other words, the radial derivative of on the circle is the radial derivative of the solution to the linear problem with an error of order .
Remark 29*.*
Considering equation (50) with gives us
[TABLE]
Since is uniformly bounded in , the above estimate implies a bound on the mean curvature of the graph, and therefore on :
[TABLE]
The graph of the solution is therefore asymptotic to a cone at infinity.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 5[5] N. V. Krylov , Lectures on elliptic and parabolic equations in Hölder spaces , vol. 12 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1996.
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