|Jun 03, 2022
We are concerned with various generalizations of the p-Laplace equation of divergence form and prove Calderon-Zygmund estimates, which implies that the integrability of the forcing term of the equation transfers to the integrability of the gradient of the solution. Firstly, we consider the Orlicz growth problem. Assuming only measurable in one of the variables and imposing small BMO in the other variables, we show the gradient estimate. This problem has many applications such as composite materials, linear laminates, and transmission problems. We will also consider the elliptic equation which has a nonlinearity with u-dependence.
Secondly, we consider double phase problems, which can be regarded as involving a non-uniformly elliptic operator. The phase changes along with the set in which the modulating coefficient a(x) is zero or positive. One of the known techniques to deal with such kind of problem is to use the difference quotient method to overcome the non-uniform ellipticity. By using extrapolation of weights, we open a new path to deal with such kind of problem. We also deal with the omega minimizers of double phase variational problems with variable exponents, and Orlicz double phase problems with variable exponents.
Finally, we consider the elliptic equations with degenerate/singular weights. We use the concept of Muckenhoupt class to deal with the weights. Imposing a small BMO condition of the logarithm of weight and Lipschitz domain with a small Lipschitz constant, we find a sharp smallness assumption along with the higher integrability of the gradient of the solution. We also introduce a result for the general weighted elliptic equation with linear growth, which has a measurable nonlinearity in one direction, and we prove the Calderon-Zygmund estimate.