Green’s Function for Second Order Elliptic Equations in Non-divergence Form
Abstract: We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has C 1,1 boundary. We also obtain pointwise bounds for the Green functions and its derivatives.
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Cited by 16 publications
References 13 publications
Estimates for Green's Functions of Elliptic Equations in Non-Divergence Form with Continuous Coefficients
We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's function for constant coefficients equations.
Estimates for Green's Functions of Elliptic Equations in Non-Divergence Form with Continuous Coefficients
We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's function for constant coefficients equations.
L2-Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type
We consider a uniformly elliptic operator $$L_A$$ L A in divergence form associated with an $$(n+1)\times (n+1)$$ ( n + 1 ) × ( n + 1 ) -matrix A with real, merely bounded, and possibly non-symmetric coefficients. If "Equation missing"then, under suitable Dini-type assumptions on $$\omega _A$$ ω A , we prove the following: if $$\mu $$ μ is a compactly supported Radon measure in $$\mathbb {R}^{n+1}$$ R n + 1 , $$n \ge 2$$ n ≥ 2 , and $$ T_\mu f(x)=\int \nabla _x\Gamma _A (x,y)f(y)\, \textrm{d}\mu (y) $$ T μ f ( x ) = ∫ ∇ x Γ A ( x , y ) f ( y ) d μ ( y ) denotes the gradient of the single layer potential associated with $$L_A$$ L A , then $$\begin{aligned} 1+ \Vert T_\mu \Vert _{L^2(\mu )\rightarrow L^2(\mu )}\approx 1+ \Vert {\mathcal {R}}_\mu \Vert _{L^2(\mu )\rightarrow L^2(\mu )}, \end{aligned}$$ 1 + ‖ T μ ‖ L 2 ( μ ) → L 2 ( μ ) ≈ 1 + ‖ R μ ‖ L 2 ( μ ) → L 2 ( μ ) , where $${\mathcal {R}}_\mu $$ R μ indicates the n-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for $${\mathcal {R}}_\mu $$ R μ , which were recently extended to $$T_\mu $$ T μ associated with $$L_A$$ L A with Hölder continuous coefficients. In particular, we show the following: If $$\mu $$ μ is an n-Ahlfors-David-regular measure on $$\mathbb {R}^{n+1}$$ R n + 1 with compact support, then $$T_\mu $$ T μ is bounded on $$L^2(\mu )$$ L 2 ( μ ) if and only if $$\mu $$ μ is uniformly n-rectifiable. Let $$E\subset \mathbb {R}^{n+1}$$ E ⊂ R n + 1 be compact and $${\mathcal {H}}^n(E)<\infty $$ H n ( E ) < ∞ . If $$T_{{\mathcal {H}}^n|_E}$$ T H n | E is bounded on $$L^2({\mathcal {H}}^n|_E)$$ L 2 ( H n | E ) , then E is n-rectifiable. If $$\mu $$ μ is a non-zero measure on $$\mathbb {R}^{n+1}$$ R n + 1 such that $$\limsup _{r\rightarrow 0}\tfrac{\mu (B(x,r))}{(2r)^n}$$ lim sup r → 0 μ ( B ( x , r ) ) ( 2 r ) n is positive and finite for $$\mu $$ μ -a.e. $$x\in \mathbb {R}^{n+1}$$ x ∈ R n + 1 and $$\liminf _{r\rightarrow 0}\tfrac{\mu (B(x,r))}{(2r)^n}$$ lim inf r → 0 μ ( B ( x , r ) ) ( 2 r ) n vanishes for $$\mu $$ μ -a.e. $$x\in \mathbb {R}^{n+1}$$ x ∈ R n + 1 , then the operator $$T_\mu $$ T μ is not bounded on $$L^2(\mu )$$ L 2 ( μ ) . Finally, we prove that if $$\mu $$ μ is a Radon measure on $${\mathbb {R}}^{n+1}$$ R n + 1 with compact support which satisfies a proper set of local conditions at the level of a ball $$B=B(x,r)\subset {\mathbb {R}}^{n+1}$$ B = B ( x , r ) ⊂ R n + 1 such that $$\mu (B)\approx r^n$$ μ ( B ) ≈ r n and r is small enough, then a significant portion of the support of $$\mu |_B$$ μ | B can be covered by a uniformly n-rectifiable set. These assumptions include a flatness condition, the $$L^2(\mu )$$ L 2 ( μ ) -boundedness of $$T_\mu $$ T μ on a large enough dilation of B, and the smallness of the mean oscillation of $$T_\mu $$ T μ at the level of B.
Given input–output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically rigorous scheme for learning the associated Green’s function G. By exploiting the hierarchical low-rank structure of G, we show that one can construct an approximant to G that converges almost surely and achieves a relative error of $$\mathcal {O}(\varGamma _\epsilon ^{-1/2}\log ^3(1/\epsilon )\epsilon )$$ O ( Γ ϵ - 1 / 2 log 3 ( 1 / ϵ ) ϵ ) using at most $$\mathcal {O}(\epsilon ^{-6}\log ^4(1/\epsilon ))$$ O ( ϵ - 6 log 4 ( 1 / ϵ ) ) input–output training pairs with high probability, for any $$0<\epsilon <1$$ 0 < ϵ < 1 . The quantity $$0<\varGamma _\epsilon \le 1$$ 0 < Γ ϵ ≤ 1 characterizes the quality of the training dataset. Along the way, we extend the randomized singular value decomposition algorithm for learning matrices to Hilbert–Schmidt operators and characterize the quality of covariance kernels for PDE learning.
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