Uniqueness of Solutions to a Nonlinear Elliptic Hessian Equation
Through an Alexandrov-Fenchel inequality, we establish the general Brunn-Minkowski inequality. Then we obtain the uniqueness of solutions to a nonlinear elliptic Hessian equation on .
According to a general Brunn-Minkowski inequality, we obtain a proof of the uniqueness of solutions to the following fully nonlinear elliptic Hessian equation: where is the support function of convex bodies, are the second-order covariant derivations of with respect to any orthonormal frame on , is the standard Kronecker symbol, is the unit sphere of -dimension, is a positive function defined on , , , and is the th elementary symmetric function defined as follows: for , The definition can be extended to any symmetric matrix by , where is the eigenvalue vector of .
Equation (1) arrives from the geometry of convex bodies. A compact convex subset of Euclidean -space with nonempty interiors is called a convex body. An important concept related to a convex body is its support function.
Definition 1. Let (the boundary of a convex body ) be a smooth, closed, uniformly convex hypersurface enclosing the origin in . Assume that is parameterized by its inverse Gauss map ; the support function of (or ) is defined by where denotes the standard inner product in .
is convex after being extended as a function of homogeneous degree in . Conversely, any continuous convex function of homogeneous degree determines a convex body as follows: From some basic concepts to support function, Minkowski sum [see Definition 4], and mixed volumes [see Definition 5], Minkowski developed a set of theories related to convex bodies. If and , (1) is the Monge-Ampère equation corresponding to the classical Minkowski problem which has been solved by Nirenberg , Pogorelov [2, 3], Cheng and Yau , and many others. When , (1) is the classical Christoffel-Minkowski problem: A necessary condition  for (6) to have a solution is where is the standard area form on . Guan et al.  obtained that (7) is sufficient for (6) to have an admissible solution [see Definition 6].
Firey  generalized the Minkowski sum to -sum [see Definition 4] from to in 1962. Later, Lutwak  extended the classical surface area measure to the -sum cases. Also in , Lutwak first introduced the general Minkowski problem, which is called -Minkowski problem thereafter. In the smooth category, -Minkowski problem is equivalent to considering the following Monge-Ampère equation: The uniqueness of -Minkowski problem for and (the uniqueness holds up to a dilation if ) has been solved in . However, the uniqueness for is difficult and still open. In , Jian et al. obtained that, for any , there exists a positive function to guarantee that (8) has two different solutions, which means that we need more conditions to consider the uniqueness.
When considering cases , attention is paid to the generalized Christoffel-Minkowski problem. In the smooth category, we need to study the -Hessian equation (1).
For (1), Hu et al.  got the existence and uniqueness of solutions to (1) when and under appropriate conditions. However, the uniqueness of (1) when has not been solved well. In this paper, we study the uniqueness of (1) for .
Our main result is the following.
Theorem 2. Suppose is a positive admissible solution of where , , , and is a positive function defined on the unit sphere and then the uniqueness holds. If , the uniqueness holds up to a dilation, which means that if solves (9), then are the whole solutions of (9).
The organization of this paper is as follows. In Section 2, we show some basic concepts and lemmas which have been obtained by Guan et al. in . In Section 3, we prove two useful propositions according to the methods in . In the last section, we prove the main theorem.
Definition 4. Given two convex bodies and in with respective support functions , , and (), the Minkowski sum is defined by the convex body whose support function is .
For , let and be two convex bodies containing the origin in in their interiors, and (). The convex body , whose support function is given by , is called Firey’s -sum of and , where “” means the -summation and “” means Firey’s multiplication.
Definition 5. Let be convex bodies in and the volume of their Minkowski sum is an th degree homogeneous polynomial of the family . Specially, the volume of is where the functions are symmetric. Then is called the Minkowski mixed volume of
Definition 7. Let be symmetric real matrices, ; the determinant of is a homogeneous polynomial of degree in . Namely, In fact, the coefficient depends only on ; then they are uniquely determined. is called the mixed discriminant of .
For later applications, we collect some results here which have been proved in .
Lemma 8. Let be the support function of convex bodies , respectively. Denoting Minkowski mixed volume by and thenwhere is the mixed discriminant [see Definition 7] of .
Remark 9. For all , setting , then where is the mixed discriminant of . Furthermore, if , denote and ; then
Lemma 10. is a symmetric multilinear form on .
Lemma 11. For any function , , , we have the Minkowski type integral formula, where is the standard area element on .
The following is a form of Alexandrov-Fenchel inequality for positive -convex functions which comes from .
Lemma 12 (Alexandrov-Fenchel inequality). If are -convex, is positive, and there exists such that on , then, for any , with equality if and only if for some constants .
3. Two Important Propositions
Now we prove two important propositions. The methods we use are from .
Proposition 13. Suppose are -convex; then with equality if and only if for some constants .
Proof. We only need to prove that is concave on . Setting , , we have By the symmetric multilinear property of , it is obvious that where the last inequality uses (20); thus is a concave function on . The equality condition is checked easily.
Proposition 14 (general Brunn-Minkowski inequality). Supposing are -convex, thenwith equality if and only if for some constants .
4. Proof of Theorem 2
Proof. Assuming that (9) has two solutions and , then we consider the equation in the following three cases.
Case 1 (). Supposing is the maximum value point of , then at , we have that is, Hence thereforethenSimilarly, we have . Thus .
Case 2 (). We have thenwhere we have used Hölder inequality in the first inequality and used (26) in the second one. Hence , which forces both the equalities to hold. By the equality condition, there exists a constant such that By (9), we know . Therefore,
Case 3 (). According to Case 2, when , we have then all the equalities hold. Thus there exists , such that . Therefore are the whole solutions of (9).
Now we complete the proof of Theorem 2.
The author declares no competing interests.
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