Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials

For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.


Introduction and main results
In this note we give a parabolic extension of a by now classical result by Kilpeläinen-Malý estimates [9], who proved pointwise estimates for solutions to quasi-linear p-Laplace type elliptic equations with measure in the right hand side, in terms of the (truncated) non-linear Wolff potential W µ β,p (x, R) of the measure, , ρ j := 2 −j ρ, j = 0, 1, 2, . . . These estimates were subsequently extended to fully nonlinear equations by Labutin [10] and fully nonlinear and subelliptic quasi-linear equations by Trudinger and Wang [17]. The pointwise estimates proved to be extremely useful in various regularity and solvability problems for quasilinear and fully nonlinear equations [9,10,14,15,17]. For the parabolic equations the corresponding result was recently given in [5,6] for the case p = 2, and by the authors in [12] for the case p > 2 and the measure on the right hand side depending on the spatial variable only. One of the main difficulties in the time dependent measure case is that of identifying the right analogue of the elliptic Wolff potential corresponding to p-Laplacian.
It is the aim of this note to introduce a parabolic version of the Wolff potential and in terms of this newly defined potential to establish pointwise estimates for solutions to parabolic equations in the degenerate case p ≥ 2 with the time-dependent measures on the right hand side. The form of the parabolic potential introduced in the note is such that it reduces to the truncated Wolff potential if the measure does not depend on time, and it reduces to the truncated Riesz potential in the case p = 2, so we recover the corresponding result in [5,6].
We are concerned with weak solutions for the divergence type quasilinear parabolic equations where Ω ⊂ R N is a domain and T > 0, and µ is an R N +1 -valued (nonnegative) Radon measure on Ω T . To this end we introduce a parabolic analog of the non-linear Wolff potentials. Before formulating the main results, let us remind the reader of the definition of a weak solution to equation (1.2).
We say that u is a weak solution to (1.2) if u ∈ V (Ω T ) := C([0, T ]; L 2 loc (Ω))∩ L p loc (0, T ; W 1,p loc (Ω)) and for any sub-domain Ω ′ ⋐ Ω and any interval I = [t 1 , t 2 ] ⊂ (0, T ) the integral identity Ω u(t)θ(t)dx The crucial role in our results is played by parabolic generalization of the truncated Wolff potential, which is defined below.
Parabolic Wolff potentials. Let µ be a positive measure on Ω T and Observe that i p (τ ) is continuous in p for every τ > 0. Also note that the above infimum is attained at some τ ∈ (0, ∞] since the function under the infimum is continuous in τ . Moreover, D 2 (ρ) = 1 2 ρ −N µ(Q ρ,ρ 2 ). Now let, for ρ > 0 and for j = 0, 1, 2, . . . set ρ j := 2 −j ρ. We define the parabolic potential for a measure µ as follows: In particular, there exists γ > 1 such that so that for p = 2 the introduced potential is equivalent to the truncated Riesz potential used in the estimates in [5,6]. Note that, for a time-independent µ charging all balls centered at x 0 , the minimum in the definition of D p (ρ) , so that in this case the introduced potential reduces to the non-linear Wolff potential. Moreover, with τ (ρ) defined as follows: it is easy to see that there exists γ = γ p > 0 such that, for all ρ > 0, and that Note that if µ is a time-independent measure then there exists γ > 1 such that The main result of this paper is the following theorem.
Remark 1.3. In case µ(dx, dt) = µ(x, t)dxdt we can estimate P |µ| p by the Lebesgue and Lorentz norms as follows.
1. Let µ ∈ L r 0, T ; L q (Ω) for r > 1 and q > N p . Then In particular, we recover a classical condition on local boundedness of the solution u (see, e.g., [3, Remark 0.1]).
By the same argument one proves that, for µ ∈ L q Ω; L r (0, T ) with r > 1 and q > N p such that 1 r + N pq < 1, the following estimate holds: 2. The latter estimates can be refined in terms of the Lorentz norms.
Recall that, for a measurable function f , the non-increasing rearrangement f * and its average f * * are defined as follows: and that the spaces L q,α , 0 < q, α ≤ ∞ are defined by the following translation-invariant metrics: It is clear that where ω N denotes the volume of a unit ball in R N . Hence The rest of the paper contains the proof of Theorem 1.1.

Proof of Theorem 1.1
We start with some auxiliary integral estimates for the solutions of (1.2) which are formulated in the next lemma. Let Note that ε p is continuous and that ε p ≥ e − 1 e > 1 2 . For λ ∈ (0, 1) we define For δ > 0 and 0 < ρ < R define, In the sequel, γ stands for a constant which depends only on N, p, c 0 , c 1 and λ, and which may vary from line to line.
To conclude (2.21) from (2.2) one has to estimate the first term in the right hand side of the latter. To do this, it suffices to observe that G(s) ≤ s and apply (2.19).