More Like this

1. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

   more » « less
2. Uniqueness of the 2D Euler equation on rough domains

   https://doi.org/10.4171/jems/1676  

   Agrawal, Siddhant; Nahmod, Andrea R (July 2025, Journal of the European Mathematical Society)

   We consider the 2D incompressible Euler equation on a bounded simply connected domain\Omega. We give sufficient conditions on the domain\Omegaso that for any initial vorticity\omega_{0} \in L^{\infty}(\Omega), the weak solutions are unique. Our sufficient condition is slightly more general than the condition that\Omegais aC^{1,\alpha}domain for some\alpha>0, with its boundary belonging toH^{3/2}(\mathbb{S}^{1}). As a corollary, we prove uniqueness forC^{1,\alpha}domains for\alpha >1/2and for convex domains which are alsoC^{1,\alpha}domains for some\alpha >0. Previously, uniqueness for general initial vorticity inL^{\infty}(\Omega)was only known forC^{1,1}domains with possibly a finite number of acute angled corners. The fundamental barrier to proving uniqueness below theC^{1,1}regularity is the fact that for less regular domains, the velocity near the boundary is no longer log-Lipschitz. We overcome this barrier by defining a new change of variable which we then use to define a novel energy functional. 

   more » « less
3. Integer Optimal Control with Fractional Perimeter Regularization

   https://doi.org/10.1007/s00245-024-10157-y  

   Antil, Harbir; Manns, Paul (July 2024, Applied Mathematics & Optimization)

   Abstract Motivated by many applications, optimal control problems with integer controls have recently received a significant attention. Some state-of-the-art work uses perimeter-regularization to derive stationarity conditions and trust-region algorithms. However, the discretization is difficult in this case because the perimeter is concentrated on a set of dimension$$d - 1$$ $d-1$for a domain of dimensiond. This article proposes a potential way to overcome this challenge by using the fractional nonlocal perimeter with fractional exponent$$0<\alpha <1$$ $0<\alpha <1$. In this way, the boundary integrals in the perimeter regularization are replaced by volume integrals. Besides establishing some non-trivial properties associated with this perimeter, a$$\Gamma $$ $\Gamma$-convergence result is derived. This result establishes convergence of minimizers of fractional perimeter-regularized problem, to the standard one, as the exponent$$\alpha $$ $\alpha$tends to 1. In addition, the stationarity results are derived and algorithmic convergence analysis is carried out for$$\alpha \in (0.5,1)$$ $\alpha \in (0.5,1)$under an additional assumption on the gradient of the reduced objective. The theoretical results are supplemented by a preliminary computational experiment. We observe that the isotropy of the total variation may be approximated by means of the fractional perimeter functional. 

   more » « less
4. Exact sampling of the infinite horizon maximum of a random walk over a nonlinear boundary

   https://doi.org/10.1017/jpr.2019.9  

   Blanchet, Jose; Dong, Jing; Liu, Zhipeng (March 2019, Journal of Applied Probability)

   Abstract We present the first algorithm that samples max n ≥0 { S n − n α }, where S n is a mean zero random walk, and n α with $$\alpha \in ({1 \over 2},1)$$ defines a nonlinear boundary. We show that our algorithm has finite expected running time. We also apply this algorithm to construct the first exact simulation method for the steady-state departure process of a GI/GI/∞ queue where the service time distribution has infinite mean. 

   more » « less
5. Degenerate linear parabolic equations in divergence form on the upper half space

   https://doi.org/10.1090/tran/8892  

   Dong, Hongjie; Phan, Tuoc; Tran, Hung (June 2023, Transactions of the American Mathematical Society)

   We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. 

   more » « less