arXiv subjects

Longjuan Xu

Publications and source records attributed to Longjuan Xu.

23 records · Page 2Linked to original sources

arXiv2019

Gradient estimates for divergence form elliptic systems arising from composite material

In this paper, we show that $W^{1,p}$ $(1\leq p<\infty)$ weak solutions to divergence form elliptic systems are Lipschitz and piecewise $C^{1}$ provided that the leading coefficients and data are of piecewise Dini mean oscillation, the lower order coefficients are bounded, and interfacial boundaries are $C^{1,\text{Dini}}$. This extends a result of Li and Nirenberg (\textit{Comm. Pure Appl. Math.} \textbf{56} (2003), 892-925). Moreover, under a stronger assumption on the piecewise $L^{1}$-mean oscillation of the leading coefficients, we derive a global weak type-(1,1) estimate with respect to $A_{1}$ Muckenhoupt weights for the elliptic systems without lower order terms.

math.AP↗

arXiv2018

Estimates for Elliptic Systems in a Narrow Region arising from Composite Materials

In this paper, we establish the pointwise upper and lower bounds of the gradients of solutions to a class of elliptic systems, including linear systems of elasticity, in a general narrow region and in all dimensions. This problem arises from the study of damage analysis of high-contrast composite materials. Our results show that the damage may initiate from the narrowest place.

math.AP↗

arXiv2018

Characterization of Electric Fields for Perfect Conductivity Problems in 3D

In composite materials, the inclusions are frequently spaced very closely. The electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large. In this paper, we establish an asymptotic formula of the electric field in the zone between two spherical inclusions with different radii in three dimensions. An explicit blowup factor relying on radii is obtained, which also involves the digamma function and Euler-Mascheroni constant, and so the role of inclusions' radii played in such blowup analysis is identified.

math.AP↗

arXiv2018

Blowup Analysis for the Perfect Conductivity Problem with convex but not strictly convex inclusions

In the perfect conductivity problem, it is interesting to study whether the electric field can become arbitrarily large or not, in a narrow region between two adjacent perfectly conducting inclusions. In this paper, we show that the relative convexity of two adjacent inclusions plays a key role in the blowup analysis of the electric field and find some new phenomena. By energy method, we prove the boundedness of the gradient of the solution if two adjacent inclusions fail to be locally relatively strictly convex, namely, if the top and bottom boundaries of the narrow region are partially "flat". The boundary estimates when an inclusion with partially "flat" boundary is close to the "flat" matrix boundary and estimates for the general elliptic equation of divergence form are also established in all dimensions.

math.AP↗

arXiv2017

Optimal estimates for the perfect conductivity problem with inclusions close to the boundary

When a convex perfectly conducting inclusion is closely spaced to the boundary of the matrix domain, a bigger convex domain containing the inclusion, the electric field can be arbitrary large. We establish both the pointwise upper bound and the lower bound of the gradient estimate for this perfect conductivity problem by using the energy method. These results give the optimal blow-up rates of electric field for conductors with arbitrary shape and in all dimensions. A particular case when a circular inclusion is close to the boundary of a circular matrix domain in dimension two is studied earlier by Ammari,Kang,Lee,Lee and Lim(2007). From the view of methodology, the technique we develop in this paper is significantly different from the previous one restricted to the circular case, which allows us further investigate the general elliptic equations with divergence form.

math.AP↗