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On the exterior Dirichlet problem for special Lagrangian equations

In this paper, we establish the existence and uniqueness theorem of the exterior Dirichlet problem for special Lagrangian equations with prescribed asymptotic behavior at infinity.


Introduction
Let D be a bounded domain in R n (n ≥ 3). We consider in this paper the Dirichlet problem for the special Lagrangian equation in the exterior domain R n \ D, where λ i (D 2 u)'s denote the eigenvalues of the Hessian matrix D 2 u, and Θ is a constant such that (n − 2)π/2 ≤ |Θ| < nπ/2. The special Lagrangian equation (1.1) originates in the calibrated geometry [HL82], and plays an important role in the study of string theory (see for example [CMMS04]). The left hand side of the equation (1.1) indeed stands for the argument of the complex number (1 + √ −1λ 1 (D 2 u))...(1 + √ −1λ n (D 2 u)), which is usually called phase or Lagrangian phase. When the phase is constant, the gradient graph {(x, Du(x))} ⊂ R n × R n = C n then is called special Lagrangian. One can prove that {(x, Du(x))} is special Lagrangian, i.e., (1.1) holds for some constant Θ ∈ (−nπ/2, nπ/2), if and only if it is a volume minimizing minimal surface in R n × R n . In the literature, the Lagrangian phase (n − 2)π/2 is usually called critical, since the level set is convex only if |Θ| ≥ (n−2)π/2 [Yuan06, Lemma 2.1]. So the main results in this paper are concerning the special Lagrangian equation (1.1) with critical and supercritical phases.
Recently, we proved in [LLY17] that any smooth solution u of (1.1) with supercritical phase in the exterior domain must tend to a quadratic polynomial Q at infinity, and satisfy u(x) = Q(x) + O 1 |x| n−2 (|x| → +∞). (1.2) That is, there exist A ∈ R n×n , b ∈ R n and c ∈ R such that lim sup |x|→+∞ |x| n−2 u(x) − 1 2 Thus for any given solution of (1.1) in the exterior domain, we know definitely that it obeys the above asymptotic behavior (1.3) at infinity. So the converse problem arises, that is, with any such prescribed asymptotic behavior at infinity, whether there exists a unique solution of the Dirichlet problem of the special Lagrangian equation (1.1) in the exterior domain? The answer is yes (at least partially), and we will mainly focus on this exterior Dirichlet problem in this paper.
Note that the solution of (1.1) is always a solution of (1.4), but the converse is not always true. Our strategy is to construct some proper subsolutions of (1.4), and then come back to show that they are exactly the desired subsolutions of (1.1). The techniques used here to construct subsolutions of (1.4) are partially inherited from our previous paper [LL16] concerning the exterior Dirichlet problem for the Hessian quotient equations, but are much harder than those and have been largely extended.
We would like to remark that, the rigidity theorems for special Lagrangian equations on the whole space have been fully studied by Prof. Y. Yuan, see for instance [Yuan02,Yuan06,WY08]; for more on the special Lagrangian equations, we refer the readers to [HL82,Fu98,CWY09,WY14] and the references therein.
For the results concerning the exterior Dirichlet problems of the Monge-Ampère equations, of the Hessian equations, and of the Hessian quotient equations, see for example [ (1) S(n) denotes the linear space of symmetric n × n real matrices; (2) Γ + denotes the positive cone Γ + := {λ ∈ R n |λ i > 0, ∀i = 1, 2, ..., n} ; (3) m(Θ, λ) is a quantity introduced by us, which plays an important role in the study of the special Lagrangian equations. For some special matrices A ∈ A Θ , for example, A = tan(Θ/n) I n , we have m(Θ, λ(A)) = n. The specific definition of m(Θ, λ) will be given in (3.32) in Subsection 3.2, when all the necessary preparation is done.
Then the main results of this paper can be stated as the following theorems.
Theorem 1.1. Let D be a bounded strictly convex domain in R n , n ≥ 3, ∂D ∈ C 2 and let ϕ ∈ C 2 (∂D). Then for any given A ∈ A Θ with (n−2)π/2 ≤ Θ < nπ/2, and any given b ∈ R n , there exists a constant c * depending only on n, D, Θ, A, b and ϕ C 2 (∂D) , such that for every c ≥ c * , there exists a unique viscosity solution u ∈ C 0 (R n \ D) of (1.7) where m ∈ (2, n] is a constant depending only on n, Θ and λ(A), which actually can be taken as m(Θ, λ(A)).
Theorem 1.2. Let D be a bounded strictly convex domain in R n , n ≥ 3, ∂D ∈ C 2 and let ϕ ∈ C 2 (∂D). Then for any given A ∈ A Θ with −nπ/2 < Θ ≤ −(n − 2)π/2, and any given b ∈ R n , there exists a constant c * depending only on n, D, Θ, A, b and ϕ C 2 (∂D) , such that for every c ≤ c * , there exists a unique viscosity solution u ∈ C 0 (R n \ D) of (1.7), where m ∈ (2, n] is a constant depending only on n, Θ and λ(A), which actually can be taken as m(−Θ, −λ(A)).
(1) By symmetry, letting u = −u, we see easily that Theorem 1.1 and Theorem 1.2 are equivalent. Thus we need only to consider a = λ(A) ∈ Γ + and prove Theorem 1.1 in the rest of this paper.
(2) A Θ is a large class of matrices containing tan(Θ/n) I n , which will be clear in Subsection 3.2. For some particular n and Θ we will see that A Θ = A 0 Θ , which is the best possible situation we can hope. For example, if n = 3, 4, Θ = ±π, we have the special Lagrangian equation in the algebraic form σ 3 (λ(D 2 u)) = σ 1 (λ(D 2 u)) (in three dimension, this indeed is det(D 2 u) = ∆u). Then the A ∈ A Θ in Theorem 1.1 and Theorem 1.2, can be chosen as any matrix A ∈ S(n) such that λ(A) ∈ L Θ ∩ (Γ + ∪ (−Γ + )), i.e., any A ∈ A 0 Θ . This result is obtained for the first time in our previous paper [LL16], since σ 3 (λ(D 2 u)) = σ 1 (λ(D 2 u)) is also a Hessian quotient equation.
(3) The restrictions on the Lagrangian phase Θ to be critical or supercritical are subtle and technical. We believe that they can be discarded more or less, since, from [Yuan02], we know that any convex solution of (1.1) (no restriction on the Lagrangian phase) in R n must be a quadratic polynomial.
(4) As far as we know, there are only two papers concerning the exterior Dirichlet problem for special Lagrangian equations, but both are highly restricted. For example, [BL13] considered the problem in the cases that n ≤ 4, Θ = ±π and A = tan(Θ/n) I n , which are just the Hessian quotient equations; [LB14] considered the problem in the cases that n ≥ 3, |Θ| > (n − 1)π/2 and A = tan(Θ/n) I n . We remark that the results appeared in [BL13] and [LB14] can all be recovered by our Theorem 1.1 and Theorem 1.2 as special cases.
The paper is organized as follows. In Section 2, we first give some specific definitions of terminologies which are standard in the literature, and a list of notations introduced only by us in this paper. Then we collect in Subsection 2.2 some well known lemmas which are mostly used in Section 4. In Section 3, we first introduce quantities Ξ k , ξ k , ξ k and investigate their properties in Subsection 3.1. Then, in Subsection 3.2, we study polynomials X, Y, Z, X, Y and Z related to the algebraic special Lagrangian equation (1.4), which play fundamental role in this paper. Section 4 is devoted to the proof of the main theorem (Theorem 1.1). To do so, we start in Subsection 4.1 to construct some appropriate subsolutions of the special Lagrangian equation (1.1), by solving some proper ordinary differential equation. Then, after reducing Theorem 1.1 to Lemma 4.3 by normalization in Subsection 4.2, we prove Lemma 4.3 in Subsection 4.3 by applying the Perron's method to the subsolutions we constructed in Subsection 4.1. The last section, Section 5, is an appendix, which is devoted to prove Lemma 3.3 asserted by us in Subsection 3.2.

Notation
In this paper, S(n) denotes the linear space of symmetric n×n real matrices, and I n denotes the identity matrix.
For A ∈ S(n) and ρ > 0, we denote by For any p ∈ R n , we write σ k (p) := 1≤s 1 <s 2 <...<s k ≤n as the k-th elementary symmetric function of p. Meanwhile, we will adopt the conventions that σ −1 (p) ≡ 0, σ 0 (p) ≡ 1 and σ k (p) ≡ 0, ∀k ≥ n + 1; and we will also define for any −1 ≤ k ≤ n and any 1 ≤ i ≤ n, and similarly for any −1 ≤ k ≤ n and any 1 ≤ i, j ≤ n, i = j, for convenience.
For the reader's convenience, we give the following list of notations which are only introduced by us in this paper.
The reasons for why we define them will be clear in future sections. We just give a few comments about them here now. To establish the existence of the solution of the special Lagrangian equation (1.1) by the Perron's method, the key point is to construct some appropriate subsolutions of its algebraic form (1.4). Since (1.4) is a fully nonlinear equation consisting of polynomials with respect to the eigenvalues of the Hessian matrix D 2 u (i.e., polynomials σ k (λ(D 2 u)), k = 1, 2, ..., n) of different order of homogeneities, to solve it we need to strike a balance among them. It will turn out to be clear that the quantities Ξ k , ξ k , ξ k and ξ k are very natural and perfectly fit for this purpose. Roughly speaking, Ξ k originates from the computation of σ k (λ(D 2 Φ(x))) where Φ(x) = φ(r A (x)) is a generalized radially symmetric function (see Lemma 2.1 and the proof of Lemma 4.2); ξ k , ξ k and ξ k result from the comparison among different σ k (λ)'s in the attempt to derive an ordinary differential equation from the original equation (1.4) (see the proof of Lemma 4.2); m(Θ, a) arises in the process of solving this ordinary differential equation (see (4.4) and (4.5) in the proof of Lemma 4.1), so do the polynomials X, Y, Z, X, Y , Z and their coefficients c k , which offer us deeper understanding of the algebraic special Lagrangian equation (1.4) and play crucial roles in the construction of the subsolutions. By Ξ k , ξ k , ξ k and ξ k , we get a good balance among different σ k (λ)'s, which can be viewed as being measured by m(Θ, a). Furthermore, we will find that m(Θ, a) has also some special meaning related to the decay and asymptotic behavior of the solution (see Lemma 4.1-(ii), Corollary 4.1 and Theorem 1.1).

Some preliminary lemmas
In this subsection, we collect some well known preliminary lemmas which will be mainly used in Section 4.
We first give a lemma to compute σ k (λ(M)) with M of certain type. If Φ(x) := φ(r) with φ ∈ C 2 , r = √ x T Ax, A ∈ S(n) ∩ Γ + and a = λ(A) (we may call Φ a generalized radially symmetric function with respect to A, according to [BLL14]), one can conclude that provided A is normalized to a diagonal matrix (see Subsection 4.2 and the proof of Lemma 4.2 for details). As far as we know, there is generally no explicit formula for λ(D 2 Φ(x)) of this type, but luckily we have a method to calculate σ k (λ(D 2 Φ(x))) for each 1 ≤ k ≤ n, which can be represented as the following lemma.
Lemma 2.1. If M = (p i δ ij + sq i q j ) n×n with p, q ∈ R n and s ∈ R, then Proof. See [BLL14].
To process information on the boundary we need the following lemma.
Lemma 2.2. Let D be a bounded strictly convex domain of R n , n ≥ 2, ∂D ∈ C 2 , ϕ ∈ C 0 (D) ∩ C 2 (∂D) and let A ∈ S(n), det A = 0. Then there exists a constant K > 0 depending only on n, diam D, the convexity of D, ϕ C 2 (D) , the C 2 norm of ∂D and the upper bound of A, such that for any ξ ∈ ∂D, there existsx(ξ) ∈ R n satisfying Proof. See [CL03] or [BLL14].
Remark 2.1. It is easy to check that Q ξ satisfy the following properties.
Let Ω ⊂ R n be a domain, ϕ ∈ C 0 (∂Ω) and let u, u ∈ C 0 (Ω) satisfying in Ω in the viscosity sense. Suppose u ≤ u in Ω, u = ϕ on ∂Ω (and additionally lim provided Ω is unbounded). Then is the unique viscosity solution of the Dirichlet problem Remark 2.2. Clearly, by letting the special Lagrangian equation (1.1) satisfies the above two lemmas.
3 Quantities and polynomials related to the special Lagrangian equations 3.1 Quantities Ξ k , ξ k , ξ k and their properties The quantities Ξ k , ξ k and ξ k originate in the calculation of σ k (λ(D 2 Φ)) and in the comparison among different σ k (λ(D 2 Φ))'s (see Lemma 2.1 and the proof of Lemma 4.2). They were partially given for the first time in our previous paper [LL16]. For completeness and convenience, we introduce them again in this subsection.
For any 0 ≤ k ≤ n and any a ∈ R n \ {0}, let and It is easy to see that Furthermore, we have the following lemma.
Lemma 3.1. Suppose a = (a 1 , a 2 , ..., a n ) with 0 < a 1 ≤ a 2 ≤ ... ≤ a n . Then (3.5) and Moreover, Proof. (1 • ) By the definitions of σ k (a) and σ k;i (a), we see that Hence we obtain (3.10) Now we show that In fact, for any i = j, similar to (3.9), we have and hence Therefore, if a i ≶ a j , then By the definition of ξ k , we have and Hence we obtain (3.14) Combining this with (3.11), (3.13) and (3.14), we deduce that Thus the proof of (3.5) is complete, and (3.8) is also clear in view of (3.12).
(2 • ) Since it follows from (3.9) that On the other hand, we have ξ n (a) = ξ n (a) = 1 which follows from Combining (3.13) and (3.9), we observe that ξ k (a) = a n σ k−1;n (a) σ k (a) = a n σ k−1;n (a) σ k;n (a) + a n σ k−1;n (a) ≤ a n σ k;n (a) σ k+1;n (a) + a n σ k;n (a) = a n σ k;n (a) where we used the inequality which is a variation of the famous Newton inequality (see [HLP34]) This completes the proof of (3.6), and similarly of (3.7).

Polynomials X, Y, Z, X, Y , Z and their properties
The polynomials appeared in (1.4) play crucial roles in this paper. For convenience, we give their names by setting for any λ ∈ R n . Write for short also We get the following elementary relations according to the trigonometry: provided the denominators are not zero. Moreover, we see that Lemma 3.2. For any λ ∈ R n , cos H(λ) and X(λ) (respectively, sin H(λ) and Y (λ)) have the same sign. That is and Proof. By (3.18), it is clear that (3.20) and (3.21) are equivalent, thus we need only to prove the former. To do this, the following assertions (which can also be used as sketches of the proof) will be adequate.
(1 • ) {λ ∈ R n | cos H(λ) = 0} has exactly n + 1 connected components, and in any two adjacent such components cos H(λ) have different signs. Here n := n(n) := n − 1, when n is odd, n, when n is even, which will be used throughout this proof and will not be mentioned again.
To confirm (1 • ), we need only to note that and there are exactly n different k ∈ Z such that Now consider (2 • ). One can easily check that X(tλ) is a polynomial with respect to t of n order. Thus it has at most n different complex roots. In particular, when λ ∈ Γ + , X(tλ) has exactly n different real roots. Indeed, since lim we see that the line {tλ} t∈R intersects with all the level surfaces L Θ (Θ ∈ (−nπ/2, nπ/2)) once and only once. This implies that there are exactly n different real t such that cos H(tλ) = 0. On the other hand, since it follows we thus conclude that the n-order polynomial X(tλ) has exactly n different real roots, which lie right in the n different level sets L Θ (Θ = π/2 + kπ ∈ (−nπ/2, nπ/2), k ∈ Z), respectively. Hence, {λ ∈ R n |X(λ) = 0} ∩ Γ + ∩ (−Γ + ) has exactly n + 1 connected components, and in any two adjacent such components X(λ) have different signs. Therefore, (2 • ) is clear.
Assertion (3 • ) is easy, thus the proof of this lemma is completed.
for any λ ∈ R n . As a corollary of Lemma 3.2, we obtain Corollary 3.1. For any a ∈ R n , we have This finishes the proof of the lemma.
We discussed in Corollary 3.1 the sign of X Y − Y X and its application. In fact, for X Y − Y X, we have established the following explicit formula, which shows immediately that X(a) Y (a) − Y (a) X(a) > 0 for all a ∈ Γ + . Lemma 3.3. For any a ∈ R n , we have Proof. See the Appendix in Section 5. (Since it is too lengthy, we attach it in the end of this paper.) The notations X, Y, X and Y we introduced above can all be viewed as "global" notations. To understand better the special Lagrangian equation (1.4), we need to introduce the following "local" notations c k , ξ k and their combinations.
For any Θ ∈ (−nπ/2, nπ/2) and any λ ∈ R n , set and We see, for any a ∈ R n , that As a corollary of Lemma 3.3, we have Proof. By Lemma 3.3, we have Since a ∈ L Θ means that H(a) = Θ, by (3.28) and Corollary 3.1, we thus deduce that The proof thereby is completed.
For any Θ ∈ (0, nπ/2) and any a ∈ L Θ ∩ Γ + , we now define . (3.32) By the proof of Corollary 3.2, we see immediately that To construct subsolutions of (1.1), we need m(Θ, λ(A)) > 2. Though a large class of matrices satisfying this, there are still some exceptions. To see this, we now give some examples.
Next we consider (1.1) in R 3 or R 4 with the Lagrangian phase Θ = π. Since A ∈ A 0 π means that a = λ(A) satisfies H(a) = π, we have sin H(a) = 0. By Lemma 3.2, we see that c 1 (a)σ 1 (a) − c 3 (a)σ 3 (a) = Y (a) = 0, Thus we obtain which implies that A π = A 0 π in three or four dimension. Finally, we give an example in R 5 to show how the distribution of the values of m(Θ, a) looks like in general. Let for any 0 < ε < π/12. Then a ε ∈ Γ + and H(a ε ) = 5π/3 ∈ (3π/2, 2π). Furthermore, we can calculate (with the help of the computer, for example) that m(ε) := m(5π/3, a ε ) = 4 √ 3 cos(4ε) + 4 √ 3 cos(2ε) + 2 √ 3 2 √ 3 cos(4ε) + 2 sin(6ε) + 2 sin(2ε) + 3 sin(4ε) . This indicates that in a wide strip around the ray {k1}, m(Θ, a) is larger than 2 in general; But when a is close to the boundary of Γ + , m(Θ, a) appears to be less than 2. We would like to mention that we have done a lot of experiments like the one above, all the results of them give us similar conclusions even when the Lagrangian phase is subcritical. For brevity, we will not add them here any more.
Now we study the properties of the polynomial Z(ta), which will play a crucial role in the construction of the subsolutions of (1.1). First, we have the following basic lemma.
Proof. The proof is elementary and hidden in the definition of c k : For the reader's convenience, we check them case by case.
By definition, it is easy to see that Z(a) = 0. This is to say that t = 1 is a root of Z(ta). The following lemma asserts that it is indeed a simple root.
Proof. Invoking the following lemma from algebra: This is evidently true by Corollary 3.2, since In fact, we can prove easily the following lemma.
Proof. Because Z(ta) cos Θ Y (ta) − sin Θ X(ta), we see that Thus it is clear that there are exactly N different t ∈ R such that −nπ/2 < H(a) = Θ + kπ < nπ/2 for some k ∈ Z, and that each such t is the root of Z(ta) = 0. Since Z(ta) is a polynomial with respect to t of N order, according to Lemma 3.4, this shows that all the roots of Z(ta) = 0 are real and simple.
We remark that, using the same method, Lemma 3.6 can be generalized as: for any fixed a ∈ Γ + , all the roots of Z(ta) = 0 are real and simple.
The positiveness of Z(ta) and its derivatives, for each t ∈ (1, +∞), is crucial in constructing the subsolutions of (1.1). This is the main reason why we restrict ourselves in the cases that the Lagrangian phases Θ are critical and supercritical. We will now discuss these problems in the following lemma.
Proof. (This simple proof was established with the help of Shanshan Ma.) Let a 1 , a 2 , ..., a n < 1 be the roots of P (x). Then there exists a constant K such that Since P (1) > 0, we know that K > 0. Thus we have Similarly, we can also obtain d k dx k P (x) > 0, ∀k = 1, 2, ..., n, ∀x ≥ 1.
We now prove Lemma 3.7.
Thus we have proved, in any case, that Z(ta) > 0, ∀t > 1. This is to say that all the roots of Z(ta) are less than or equal to 1. Invoking Lemma 3.6 and by a slightly modified version of Lemma 3.8, (i.e., we need to consider 1 + ǫ for any small ǫ > 0 rather than 1 in the deducing of d dt (Z(ta)) > 0, ∀t > 1; as for d dt (Z(ta))| t=1 > 0, it has already been proved in Corollary 3.2, or actually can be viewed as a corollary of Lemma 3.5 or Lemma 3.6. Once d dt (Z(ta)) > 0 (∀t ≥ 1) is established, the higher order derivative inequalities can be obtained directly by Lemma 3.8.) the rest assertions of this lemma are clear.
Remark 3.1. To see Z(ta) > 0 (∀t > 1) quickly, there is an easy point of view which will be stated as below, although we still believe that the above proof are of more value. By Lemma 3.4 and the proof of Lemma 3.6, we see that t is a root of the N-order polynomial Z(ta) if and only if H(ta) = Θ+kπ.
Since we now take Θ ∈ (n−2)π/2, nπ/2 , there are exactly N different such roots t and t = 1 is the largest one. By Lemma 3.4, we know that the leading coefficient c N of Z(ta) is positive, thus Z(ta) > 0 (∀t > 1) is evident.
Proof. By (3.28) and Lemma 3.7, it is clear that

Proof of the main theorem 4.1 Construction of the subsolutions
In this subsection, we prove the following key lemma and then use it to construct subsolutions of the special Lagrangian equation (1.1). Note that for the generalized radially symmetric subsolution Φ(x) = φ(r) that we want to construct, the solution ψ(r) discussed in the following lemma actually is corresponding to φ ′ (r)/r, informally those major eigenvalues of the Hessian D 2 Φ (see the proof of the Lemma 4.2).
Proof. For simplicity of notation, we will often write ψ(r) or ψ(r, β) (respectively, c k (Θ), ξ k (Θ, a), σ k (a)) simply as ψ (respectively, c k , ξ k , σ k ), when there is no confusion. The proof of this lemma now will be divided into three steps.
Step 1. In this step, we largely follow the strategy of the one in our previous paper [LL16], although the foundations of the statements are totally different.
We deduce from (4.1) that where we set  If β = 1, then ψ(r) ≡ 1 is a solution of the problem (4.3), since g(1) = 0 according to Lemma 3.7. Thus, by the uniqueness theorem for the solution of the ordinary differential equation, we know that ψ(r, 1) ≡ 1 is the unique solution satisfies the problem (4.3).
Thus ψ(r) = ψ(r, β) is strictly decreasing with respect to r, this excludes the case (2 • ) above. We assert that the case (3 • ) can also be excluded. Otherwise, the solution curve must intersect with {ν = 1} at some point (r 0 , ψ(r 0 )) on it and then tends to {ν = ν 0 } after crossing it. But ψ(r) ≡ 1 is also a solution through (r 0 , ψ(r 0 )) which contradicts the uniqueness theorem for the solution of the initial value problem of the ordinary differential equation.
Thus we complete the proof of the existence and uniqueness of the solution ψ(r) = ψ(r, β) of the problem (4.1) on [1, +∞).
Letting x = 1, we conclude that , which gives formula to calculate K 1 .
In particular, we see that ψ(r, β) is strictly increasing with respect to β.
Integrating it from M to β k and recalling ψ(1, β k ) = β k , we get which is a contradiction. Hence the assertion (iii) of the lemma is proved, and we thus complete the proof of the whole lemma. Set Note that the integral on the right hand side is convergent, in view of Lemma 4.1-(ii). Moreover, as an application of Lemma 4.1, we have the following corollary.
Since Z(t1) is a polynomial of N order, which has exactly N real and simple roots, and approaches to +∞ as t → +∞, we conclude that Combing (4.15), (4.16) with Z(λ(D 2 Φ)) ≥ 0 we derived above, by the continuity of Φ, we obtain This completes the proof of Lemma 4.2.

Lemma 4.3 implies Theorem 1.1
In this subsection, we introduce the following lemma which is a special and simple case of Theorem 1.1 with the additional condition that the matrix A is diagonal and the vector b vanishes.
Lemma 4.3. Let D be a bounded strictly convex domain in R n , n ≥ 3, ∂D ∈ C 2 and let ϕ ∈ C 2 (∂D). Then for any given diagonal matrix A ∈ A Θ with (n − 2)π/2 ≤ Θ < nπ/2, there exists a constant c * depending only on n, D, Θ, A and ϕ C 2 (∂D) , such that for every c ≥ c * , there exists a unique viscosity solution where m = m(Θ, λ(A)) ∈ (2, n]. To prove Theorem 1.1, it suffices to prove Lemma 4.3. Indeed, suppose that D, ϕ, A and b satisfy the hypothesis of Theorem 1.1. Consider the decomposition A = Q T ΛQ, where Q is an orthogonal matrix and Λ is a diagonal matrix which satisfies λ(Λ) = λ(A). Let By Lemma 4.3, we conclude that there exists a constant c * depending only on n, D, Θ, Λ and φ C 2 (∂ D) , such that for every c ≥ c * , there exists a unique where m = m(Θ, λ(Λ)) = m(Θ, λ(A)) ∈ (2, n]. Let We assert that u is the solution of (1.7) in Theorem 1.1. To show this, we need only to note that and Thus we have proved that Theorem 1.1 can be established by Lemma 4.3.

Proof of Lemma 4.3
Now we use the Perron's method to prove Lemma 4.3. The procedures employed here are standard and quite similar to the ones in our previous paper [LL16]. For the reader's convenience, we give the full details.
(6) As showed in (4.11) and (4.12), for any β ≥ 1, we have where we set and used the fact that Step 2.  On the other hand, it follows from Corollary 4.1 that µ(β) is continuous and strictly increasing with respect to β which indicates that the inverse of µ(β) exists and µ −1 is strictly increasing . Thus there exists a unique β(c) such that µ(β(c)) = c. Then we have Invoking the monotonicity of Φ β with respect to β and (4.20), we obtain Φ β(c) ≥ Φβ > Q on ∂Er. (4.21) Note that we already know Φ β(c) ≤ Q on Er \ D, from (4) of Step 1. Let Then we have (1) u is continuous and satisfies in R n \ D in the viscosity sense, by (1) and (3) of Step 1.
To use the Perron's method to establish Lemma 4.3, we now need only to prove that u ≤ u in R n \ D.
In fact, since On the other hand, for every ξ ∈ ∂D, since follows from (4.21) and (4.22), we obtain .
we deduce from the comparison principle that Hence Q ≤ u in Er \ D.
Recall that we have defined Proof of Lemma 5.1. The proof will be divided into four steps.
Step 1. We have if n is even, (5.3) and the last equality follows from the rearrangement (see for example Figure  1) realized by + 1), 2i = p + q + 1 and 2j + 1 = p − q.
Note that the p here, unlike the n in (5.3), is independent of the parity (oddness or evenness) of the dimension n.
The calculation of (5.6) will be given in Step 4. To do this, we need some other preparations which will be presented in the following two steps.
Step 2. We now prove that For Q = 0, (5.7) is obviously true. So we may assume that Q ≥ 1. Then where we have used the formulas In order to complete the proof of (5.7), we must prove (5.8) and (5.9). To verify assertion (5.8), we check directly that To prove assertion (5.9), we first note that this is equivalent to Next, to show (5.10), we need only to note that where the last equality holds since Thus the proof of (5.7) is complete.
Note that S 0 k = σ k and S j k (1) = C k n C j k , where 1 := (1, 1, ..., 1) ∈ R n , which means that S j k is a polynomial of C k n C j k terms and can be viewed as a generalization of the elementary symmetric polynomial σ k .
To do this, we first observe that σ j (a)σ k (a) is a polynomial with terms of the S M L (a) types, where 0 ≤ L ≤ M ≤ n and L + M = j + k. Thus for j + k ≤ n, any term of σ j σ k can only be one of the elements of Hence it remains to determine the coefficient of each S M L (a). To do so, we first recognize that, for any fixed a = (a 1 , a 2 , ..., a n ) ∈ R n , there are N terms Thus the proof of (5.11) and (5.12) is complete.