Equations with Lipschitz Nonlinearities

Abstract

We start here by studying the porous media equation problem (1.1) when $\beta: \mathbb{R} \rightarrow \mathbb{R}$ is monotonically increasing and Lipschitz continuous. The main reason is that general maximal monotone graphs β can be approximated by their Yosida approximations β _ε which are Lipschitz continuous and monotonically increasing. So, several estimates proved in this chapter will be exploited later for studying problems with more general β.

Appendix: Two Analytical Inequalities

Let us consider the Laplace operator in $L^{2}(\mathcal{O}),\;\mathcal{O}\in \mathbb{R}^{d},$ with homogeneous boundary conditions and its orthonormal basis of eigenfunctions, that is

$$\displaystyle{ -\varDelta e_{k} =\alpha _{k}e_{k}\;\mbox{ in}\;\mathcal{O},\quad e_{k} = 0\;\mbox{ on}\;\partial \mathcal{O}. }$$

(2.82)

We set f _k  = α _k ^1∕2 e _k so that {f _k } is an orthonormal basis in $H^{-1}(\mathcal{O})$. We assume that $\partial \mathcal{O}$ is sufficiently regular (for instance of class C ₂) in order to apply [66].

Proposition 1

There exist C ₁ > 0 and C ₂ > 0 such that

$$\displaystyle{ \|xf_{k}\|_{-1} \leq C_{1}\alpha _{k}^{d}\|x\|_{ -1}^{2},\quad \forall \;k \in \mathbb{N}. }$$

(2.83)

and

$$\displaystyle{ \vert xe_{k}\vert _{2}^{2} \leq C_{ 2}\alpha _{k}^{d-1}\vert x\vert _{ 2}^{2},\quad \forall \;k \in \mathbb{N}, }$$

(2.84)

Proof

The proof of (2.84) is very simple. In fact for each x ∈ L ² we have

$$\displaystyle{\vert xe_{k}\vert _{2} \leq \vert x\vert _{2}\,\vert e_{k}\vert _{\infty }\leq c\alpha _{k}^{\frac{d-1} {2} }\vert x\vert _{2},\quad \forall \;k \in \mathbb{N},}$$

because by [66] we have $\vert e_{k}\vert _{\infty }\leq c\alpha _{k}^{\frac{d-1} {2} }$ for all $k \in \mathbb{N}$.

Let us now consider (2.83). Since H ⁻¹ is the dual of H ₀ ¹ we have

$$\displaystyle{ \vert xe_{k}\vert _{-1}^{2} =\sup \left \{\vert \langle xe_{ k},\varphi \rangle \vert _{2}^{2}:\;\varphi \in H_{ 0}^{1},\;\vert \varphi \vert _{ H_{0}^{1}} \leq 1\right \}. }$$

(2.85)

But

$$\displaystyle{\vert \langle xe_{k},\varphi \rangle \vert _{2}^{2} = \vert \langle x,e_{ k}\varphi \rangle \vert _{2}^{2} \leq \vert x\vert _{ -1}^{2}\vert e_{ k}\varphi \vert _{H_{0}^{1}}^{2}}$$

On the other hand, for all $k \in \mathbb{N}$

$$\displaystyle{\begin{array}{lllll} \vert e_{k}\varphi \vert _{H_{0}^{1}}^{2} & =&\vert \nabla (e_{ k}\varphi )\vert _{2}^{2} & =& -\int _{ \mathcal{O}}e_{k}\,\varphi \,\varDelta (e_{k}\,\varphi )\,d\xi \\ \\ & & & =& -\int _{\mathcal{O}}(e_{k}\,\varphi ^{2}\,\varDelta e_{ k} + e_{k}^{2}\varphi \,\varDelta \varphi + \tfrac{1} {2}\nabla (e_{k}^{2}) \cdot \nabla (\varphi ^{2}))d\xi \\ \\ & & & =& -\int _{\mathcal{O}}(e_{k}\,\varphi ^{2}\,\varDelta e_{k} + e_{k}^{2}\varphi \,\varDelta \varphi -\tfrac{1} {2}\,e_{k}^{2}\,\varDelta (\varphi ^{2}))d\xi \end{array} }$$

Since

$$\displaystyle{\varDelta (\varphi ^{2}) = 2\varphi \;\varDelta \varphi + 2\vert \nabla \varphi \vert ^{2},}$$

we have

$$\displaystyle{ \vert e_{k}\varphi \vert _{H_{0}^{1}}^{2} =\int _{ \mathcal{O}}(\alpha _{k}\varphi ^{2} + \vert \nabla \varphi \vert ^{2}))e_{ k}^{2}d\xi,\quad \forall \;k \in \mathbb{N}. }$$

(2.86)

Therefore,

$$\displaystyle{ \vert e_{k}\varphi \vert _{H_{0}^{1}}^{2} \leq \alpha _{ k}\vert \varphi e_{k}\vert _{2}^{2} + \vert \varphi \vert _{ H_{0}^{1}}^{2}\vert e_{ k}\vert _{\infty }^{2},\quad \forall \;k \in \mathbb{N}. }$$

(2.87)

Now by the Sobolev embedding theorem we have $H_{0}^{1} \subset L^{ \frac{2d} {d-2} }$ for d > 3, H ₀ ¹ ⊂ ∩ _p ≥ 2 L ^p for d = 1, 2 with continuous embedding. Then, using Hölder in the first term of (2.87) we see that there is a constant c > 0 such that

$$\displaystyle{ \vert e_{k}\varphi \vert _{H_{0}^{1}}^{2} \leq (c\alpha _{ k}\vert e_{k}\vert _{d}^{2} + \vert e_{ k}\vert _{\infty }^{2})\vert \varphi \vert _{ H_{0}^{1}(\mathcal{O})}^{2}. }$$

(2.88)

Now as mentioned earlier we know that

$$\displaystyle{\vert e_{k}\vert _{\infty }^{2} \leq c_{ 1}\alpha _{k}^{d-1},\quad \forall \;k \in \mathbb{N},}$$

so that, by interpolation³

$$\displaystyle{\vert e_{k}\vert _{d}^{2} \leq c_{ 2}\alpha _{k}^{\frac{(d-1)(d-2)} {d} },\quad \forall \;k \in \mathbb{N},}$$

Finally, we find

$$\displaystyle{ \vert e_{k}\varphi \vert _{H_{0}^{1}}^{2} \leq c(\alpha _{ k}^{1+\frac{(d-1)(d-2)} {d} } +\alpha _{ k}^{d-1})\vert \varphi \vert _{H_{ 0}^{1}}^{2} \leq c_{ 1}\alpha _{k}^{d-1}\vert \varphi \vert _{ H_{0}^{1}}^{2},\quad \forall \;k \in \mathbb{N}, }$$

(2.89)

and therefore by (2.85) (2.83) follows.