1,721,222 research outputs found
No Lavrentiev gap for some double phase integrals
No full textWe prove the absence of the Lavrentiev gap for non-autonomous functionals F(u) := integral(Omega) f(x, Du(x)) dx,where the density f(x, z) is alpha-Holder continuous with respect to x; x is an element of Omega subset of R^n; f satisfies the (p, q)-growth conditions |z|^p <= f(x, z) <= L (1 + |z|^q),where 1 < p < q < p(n+alpha)/n, and it can be approximated from below by suitable densities f_k.We prove the absence of the Lavrentiev gap for non-autonomous functionalsF(u) := integral(Omega) f(x, Du(x)) dx,where the density f(x, z) is alpha-Holder continuous with respect to x is an element of Omega subset of R-n, it satisfies the (p, q)-growth conditionsvertical bar z vertical bar(p) <= f(x, z) <= L (1 + vertical bar z vertical bar(q)),where 1 < p < q < p(n+alpha/n), and it can be approximated from below by suitable densities f(k)
Quasiconformal solutions to certain first order systems and the proof of a conjecture of G. W. Milton.
No full textAbstractWe study fine properties of the gradient of the solution to the conductivity equation div (σ(χ)▿u(χ)) = 0 in bounded domains. Our analysis is restricted to dimension two and it is concerned with merely measurable, elliptic coefficients. We establish sharp results on the higher order of integrability of the modulus of the gradient and of the inverse of the modulus of the gradient of the solution with the aid of recent advances in the theory of quasiconformal mappings due to Astala, Eremenko and Hamilton.We also consider the first order system associated to the second order elliptic equation, hence defining the map w = (u, ũ) and we isolate a class of Dirichlet boundary data on the function u which guarantees the quasiconformality of the mapping w. This leads in particular to a geometrical characterization of the electrostatic energy. We make use of results about the critical points of solutions of elliptic equations due to Alessandrini and Alessandrini and Magnanini
Differentiability for bounded minimizers of some anisotropic integrals
Full text linkWe prove the existence of second weak derivatives for bounded minimizers u: Ω ⊂ Rn → RN of the integral ∫Ω(|Du|2 + |Dnu|q) dx, when 2 < q ≤ 2(n − 1)/(n − 3), n ≥ 4. This allows us to improve on the Hausdorff dimension of the singular set of u
Local boundedness for solutions of a class of nonlinear elliptic systems
Full text linkIn this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of m equations in divergence form, satisfying p growth from below and q growth from above, with p≤ q; this case is known as p, q-growth conditions. Well known counterexamples, even in the simpler case p= q, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component uα of the solution u= (u1,.. , um) satisfies an improved Caccioppoli’s inequality and we get the boundedness of uα by applying De Giorgi’s iteration method, provided the two exponents p and q are not too far apart. Let us remark that, in dimension n= 3 and when p= q, our result works for 3/
On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in R^2. preprint, arXiv:1405.1541
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