We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in

The knowledge of the data makes all the difference in the real-world applications of boundary value problems. Quantitative estimates are of extreme importance in any other area of science such as engineering, biology, geology, and even physics, to mention a few. In the existence theory to the nonlinear elliptic equations, fixed point arguments play a crucial role. The solution may exist such that it is estimated in an appropriate functional space, where the boundedness constant is frequently given in an abstract way. Their derivation is so complicated that it is difficult to express them, or they include unknown ones that are achieved by a contradiction proof, as, for instance, the Poincaré constant for nonconvex domains. The majority of works consider the same symbol for any constant that varies from line to line along the whole paper (also known as universal constant). In conclusion, the final constant of the boundedness appears completely unknown from the physical point of view. In presence of this, our first concern is to exhibit the dependence on the data of the boundedness constant. To this end, first (Section

Dirichlet, Neumann, and mixed problems with respect to uniformly elliptic equation in divergence form are widely investigated in the literature (see [

It is known that the information that “The gradient of a quantity belongs to a

Let

Let us consider the following boundary value problem, in the sense of distributions:

Set for any

Defining the

For

One says that

Since

The presented results in this section are valid whether

We recall the existence result in the Hilbert space

If

For

If

Taking

For

Consider the case of dimension

If

The existence and uniqueness of a weak solution

The meaning of the Neumann solution

The existence of a solution is recalled in the following proposition in accordance with

Let

For each

In order to pass to limit (

By the Hölder inequality with exponents

Let us choose

For

For both cases, we can extract a subsequence of

In terms of Proposition

The existence of a solution, which is given at Proposition

Finally, we state the following version of Proposition

Let

Since the Dirac delta function

In this section, we establish some maximum principles, by recourse to the Stampacchia technique [

Let

Let

Making use of (

This completes the proof of Proposition

The Dirichlet problem studied by Stampacchia in [

Let us extend Proposition

Under the conditions of Proposition

Let

Therefore, we conclude

Applying (

Finally, we find (

Next, let us state the explicit local estimates. The Caccioppoli inequality (

Let

Fix

Making use of (

Applying the properties of

In order to apply [

Therefore, the proof of Proposition

The cut-off function explicitly given in Proposition

Let us prove the corresponding Neumann version of Proposition

Let

Fix

Applying the properties of

The set

2D schematic representations of a Lipschitz domain

Finally, we state the following local version that will be required in Section

Let

First we argue as in Proposition

In this section, we reformulate some properties of the Green kernels.

For each

The existence of the Green function

Let

For any

In order for

In order to prove the nonnegativeness assertion, first calculate

For each

Hence, using (

Since

For each

For each

Since our concern is on weak solutions to (

Let

For each

Hereafter,

Let

For each

To prove estimate (

Notice that

Next, we prove additional estimates for the derivative of the weak solution to (

Let

By density, since

Fix

In order to determine the final constant in (

Considering that, for all

Let us analyze the first integral of the right-hand side in (

Returning to (

In an

Observing (

The upper bound in (

Let

Let

Let

The regularity theory for solutions of the class of divergence form elliptic equations in convex domains guarantees the existence of a unique strong solution if the coefficient is uniformly continuous, taking the Korn perturbation method [

For a domain of class

For Lipschitz domains with small Lipschitz constant, the Neumann problem is solved in [

Let us begin by establishing the relation between any weak solution

For every

In the presence of the Hardy-Littlewood-Sobolev inequality, we prove the following

Let

Since

For the particular situation, we choose

Having the results established in Section

Let

Differentiating (

Let

Let us estimate the last integral on the right-hand side in (

Finally, inserting the above inequality into (

The author declares that there is no conflict of interests regarding the publication of this paper.