In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e., every eigenfunction of the second-smallest e...In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian(introduced by Kigami) attains its maximum and minimum on the boundary.展开更多
In this paper, we define a class of domains in R^n. Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary cond...In this paper, we define a class of domains in R^n. Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary conditions. We then show that the hot spots conjecture holds for this class of domains.展开更多
We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motiva...We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincar′e inequalities, Harnack inequalities, and NTA(non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.展开更多
In the paper: the representation of large ev en integer as a sum of two primes is proved to be right independently by each of W-progression ∑(∞)(X n=1D)(n+1)(n-1)!of the discovery and the prime theorem. I t is induc...In the paper: the representation of large ev en integer as a sum of two primes is proved to be right independently by each of W-progression ∑(∞)(X n=1D)(n+1)(n-1)!of the discovery and the prime theorem. I t is induced as two following problems which are solved for getting results of ration: Is there a function of f(2n) to be only depend ent upon 2n or not? And it can express a number of group of prime solutions on r epresentatio n of even integer as a sum of two primes. In one－ dimensional space, the prime t heorem is led into odd sequence integer to find P(G)～2 log n is regarded as a data handling tool for setting a mathematical model of ran dom sampling, get:P2n(1，1)n＞2 2n-P 2=P 1=f(2n)～(2nlogn/2log2nlog2n(2n→∞). The prime theorem π(x) is gene ralized to the two-dimensional space: π(x,y). A mathematical model of average values is set up by π(x,y), get: P2n(1，1)2 (X n＞2 2n=P 1+P 2)=f(2n)2～(2n log22n SX) (2n→∞). But for expressing a number of group of prime solutions of even integer，the laws of values of principal steps of the two different functions f(2n) and f(2n) 2 are unanimous. Thus, the proof of different ways lead to the same result and determines a forceful declaration: Goldbach’s conjecture is proved to be a right theorem.展开更多
基金supported in part by NSFC grants Nos.11271327, 11771391
文摘In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian(introduced by Kigami) attains its maximum and minimum on the boundary.
文摘In this paper, we define a class of domains in R^n. Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary conditions. We then show that the hot spots conjecture holds for this class of domains.
基金Supported in part by NSF(Grant No.DMS 1500771)a Simons Fellowshipthe Simons Foundation(Grant No.601948 DJ)
文摘We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincar′e inequalities, Harnack inequalities, and NTA(non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.
文摘In the paper: the representation of large ev en integer as a sum of two primes is proved to be right independently by each of W-progression ∑(∞)(X n=1D)(n+1)(n-1)!of the discovery and the prime theorem. I t is induced as two following problems which are solved for getting results of ration: Is there a function of f(2n) to be only depend ent upon 2n or not? And it can express a number of group of prime solutions on r epresentatio n of even integer as a sum of two primes. In one－ dimensional space, the prime t heorem is led into odd sequence integer to find P(G)～2 log n is regarded as a data handling tool for setting a mathematical model of ran dom sampling, get:P2n(1，1)n＞2 2n-P 2=P 1=f(2n)～(2nlogn/2log2nlog2n(2n→∞). The prime theorem π(x) is gene ralized to the two-dimensional space: π(x,y). A mathematical model of average values is set up by π(x,y), get: P2n(1，1)2 (X n＞2 2n=P 1+P 2)=f(2n)2～(2n log22n SX) (2n→∞). But for expressing a number of group of prime solutions of even integer，the laws of values of principal steps of the two different functions f(2n) and f(2n) 2 are unanimous. Thus, the proof of different ways lead to the same result and determines a forceful declaration: Goldbach’s conjecture is proved to be a right theorem.
