$L^p$-estimates and regularity for SPDEs with monotone semilinearity

Semilinear stochastic partial differential equations on bounded domains $\mathscr{D}$ are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. A typical example is the stochastic Ginzburg--Landau equation. The main result of this article are $L^p$-estimates for such equations. The $L^p$-estimates are subsequently employed in obtaining higher regularity. It is shown, under appropriate assumptions, that the solution is continuous in time with values in the Sobolev space $H^2(\mathscr{D}')$ and $L^2$-integrable with values in $H^3(\mathscr{D}')$, for any compact $\mathscr{D}' \subset \mathscr{D}$. Using results from $L^p$-theory of SPDEs obtained by Kim we get analogous results in weighted Sobolev spaces on the whole $\mathscr{D}$. Finally it is shown that the solution is H\"older continuous in time of order $\frac{1}{2} - \frac{2}{q}$ as a process with values in a weighted $L^q$-space, where $q$ arises from the integrability assumptions imposed on the initial condition and forcing terms.


INTRODUCTION
The aim of this article is to obtain L p -estimates and regularity of solutions to the semilinear stochastic partial differential equation (SPDE) where, Here D is a bounded domain in R d and W k are independent Wiener processes. The coefficients a and σ are assumed to satisfy stochastic parabolicity condition (and thus our equation is non-degenerate). Moreover all the coefficients a, b, c, σ and µ are assumed to be measurable and bounded, f = f t (ω, x, r, z) is measurable, continuous in (r, z), monotone in r except perhaps around the origin, Lipschitz continuous in z, bounded in x and of polynomial growth in r (of arbitrary order). The forcing terms f 0 and g are assumed to satisfy appropriate integrability conditions. A typical example of equation fitting this setting is the stochastic Ginzburg-Landau equation. In this case f (r) = −|r| α−2 r , α ≥ 1.
To obtain higher interior regularity we will have to impose further regularity assumptions on the coefficients. To obtain regularity up to the boundary (in weighted Sobolev spaces) we will also need to impose regularity assumptions on the domain. The assumptions will be formulated precisely in further sections.
The main aim of this article is to obtain regularity results for the solutions to the SPDE (1). For a semilinear equation it is natural to consider the term f := f (u, ∇u) + f 0 as a free term in an appropriate linear SPDE and to use established methods and theory to obtain regularity for this linear SPDE. Due to uniqueness of solutions to (1), see Lemma 1, we then get the same regularity for the semilinear equation (1). However, for the theory of regularity of linear SPDEs to apply, we need that the new free term f satisfies appropriate integrability conditions. This would typically mean at least L 2 -integrability. Since the semilinear term in (1) is allowed arbitrary polynomial growth, it is clear that we need to obtain L p -estimates for solution to (1) with p ≥ 2 sufficiently large. Note that if one attempts to do this using Sobolev embedding theorem then one immediately runs into restrictions on the combination of dimension of D and the growth of the semilinear term. The main novelty of this article is in allowing arbitrary dimension of D and growth of the semilinear term. See Theorem 1. This is achieved by using the monotonicity property of the semilinear term and a cutting argument to obtain the required L p -estimate. Once these have been established we then obtain new spatial regularity results for the SPDE (1), these are both interior regularity and up-to-the-boundary regularity in weighed Sobolev spaces. See Theorems 2 and 5. Finally we have a new time regularity result (in weighted space again), see Theorem 6. Regularity of solutions to linear SPDEs has been an area of active interest for quite some time and here we point out some of the main results. Regularity of solutions to linear SPDEs on the whole space has been proved in Rozovskii [19]. On domains with a boundary the situation is much more involved and one cannot expect the same regularity up to the boundary as in the interior of the domain. See e.g. Examples 1.1 and 1.2 in Krylov [15]. After this observation two approaches to dealing with boundaries emerge: one is to quantify the loss of regularity near the boundary using weighted Sobolev spaces. These allow oscillations and explosion of the spatial derivatives of the solution near the boundary. The other approach is to side-step the problems created by the boundary by restricting the class of equations under consideration by imposing additional restriction on the noise term near the boundary (effectively disallowing stochastic forcing near the boundary). See Flandoli [3]. Weighted Sobolev spaces have also further employed, in the context of L p -thoery for linear SPDEs, by Kim [12].
Unsurprisingly, there are fewer results for nonlinear SPDEs. Kim and Kim use the L ptheory in [10] and [11] to obtain regularity for quasilinear SPDEs where the coefficients are uniformly bounded. Current results in Gerencsér [7] show that for a class of SPDEs, including (1), there exists some Hölder exponent such that the solution is Hölder continuous in space up to the boundary with this exponent. For interior regularity of a class of quasilinear equations associated with the "p-Laplace" operator see Breit [1]. For SPDEs with drift given by the subgradient of a quasi-convex function and with sufficiently regular noise Gess [4] proves higher regularity and existence of (analytically) strong solutions. All the aforementioned work on regularity of nonlinear SPDEs has been done using the variational approach. For results obtained in the semigroup framework we refer the reader to the work of Jentzen and Röckner [5] and references therein.
The article is organised as follows: Section 2 is devoted to the proof of Theorem 1 which gives us the desired L p -estimates for the solution to semilinear SPDE (1). In Section 3, we first prove interior regularity for the associated linear SPDE, see Theorem 3. We then use the results on interior regularity of the linear SPDE to prove Theorem 2. In Section 4, we prove regularity results up to the boundary and time regularity in weighted Sobolev spaces using L p -theory from Kim [12]. The main results and required assumptions are stated at the beginning of each section.

L p -ESTIMATES FOR THE SEMILINEAR EQUATION
Let T > 0 be given, (Ω, F , (F t ) t∈[0,T ] , P) be a stochastic basis, P be the predictable σ-algebra and W := (W t ) t∈[0,T ] be an infinite dimensional Wiener martingale with respect to Further, let D be a bounded domain in R d with Lipschitz boundary. We use standard notation for Lebesgue-Bochner and Sobolev spaces. In general, if X is a normed linear space then we will use | · | X to denote the norm in this space. There are exceptions: if x ∈ R d then |x| denotes the Euclidean norm. For Lebesgue and Sobolev spaces over the entire domain D we will omit the dependence on D. So e.g. if h ∈ L p (D) then we will write |h| L p for |h| L p (D) . If h ∈ L p ((0, T ); L p (D)) then we use h L p to denote the norm. Throughout this article N denotes a generic constant that may change from line to line.

Remark 1.
Without loss of generality, we may assume that almost surely for all t, x and z the function r → f t (x, r, z) is decreasing. If not, then (1) can be rewritten by replacing f t (x, r, z) withf t (x, r, z) := f t (x, r, z) − Kr and c t (x) withc t (x) := c t (x) + K, where using Assumption A -3,f is decreasing in r. Further, we may assume that almost surely for all t and x, f t (x, 0, 0) = 0. Otherwise, we can replace f t (x, r, z) in (1) byf t (x, r, z) := f t (x, r, z) − f t (x, 0, 0) and f 0 t bỹ f 0 t (x) := f 0 t (x) + f t (x, 0, 0). Definition 1 (L 2 -Solution). An adapted, continuous L 2 (D)-valued process is said to be a solution of stochastic partial differential equation (1) if (ii) almost surely for every t ∈ [0, T ] and ξ ∈ C ∞ 0 (D), The following theorem is the main result of this section.
The rest of Section 2 is devoted to proving Theorem 1 but we give a brief outline of the proof here.
(1) We replace the semilinear term f by truncations f m , depending on some m ∈ N, chosen in such a way that that the monotonicity is preserved and f m are bounded. For standard theory of stochastic evolution equations we obtain u m which are solutions to the SPDE with f replaced with f m . (2) We now wish to get the estimate (3) for these u m (uniformly in m). If we were allowed to apply Itô's formula directly to r → |r| p and the process u m t (x) and to integrate over D then (3) for u m would follow from A-1, A-2 and A-3.
(3) Since, of course, this is not allowed we instead consider an appropriate bounded smooth approximation φ n to r → |r| p and use the Itô formula from Krylov [14]. We then establish an estimate similar to (3) but for φ n (u m ) instead of |u m | p and with the right-hand-side still depending on m but independent of n. See Lemma 2. This allows us to take the limit n → ∞ and to use the monotonicity of r → f m t (x, r, z) to obtain (3) for u m . See Lemma 3. (4) The final step is then to use compactness argument to obtain u as a weak limit of (u m ) m∈N , see Lemma 4, and the usual monotonicity argument to show that u satisfies (1). Fatou's lemma will then yield (3) for u. Before proceeding with the proof of Theorem 1, we observe the following: Remark 2. Assumptions A-1 and A-2 imply, after some computations using Hölder's and Young's inequalities, the existence of a constant K ′ depending on K, d and κ only such that almost surely for all t ∈ [0, T ] and w, w ′ ∈ H 1 0 (D), Lemma 1 (Uniqueness). The solution to (1) is unique in the sense that if u andū both satisfy (1) then Proof. Let u andū be two solutions of (1) in the sense of Definition 1. Then, almost surely for all t ∈ [0, T ]. Using Remark 1, Assumption A-3 and Young's inequality, we get Using the product rule and applying Itô's formula for the the square of the norm to (4), see Gyöngy andŠiška [9] or Pardoux [18, Chapitre 2, Theoreme 5.2], we obtain almost surely for all t ∈ [0, T ]. Substituting (5) in (6) and using Remark 2, we get implying that right hand side is a non-negative local martingale (and thus a super-martingale) starting from 0 and hence for all t ∈ [0, T ], Having proved uniqueness we start preparing the proof of Theorem 1. For m ∈ N, consider the truncated function and the equation For each m ∈ N, using Assumption A-3, f m t (x, r, z) is bounded and hence (7) can be viewed as a SPDE on the Gelfand triple H 1 0 (D) ֒→ L 2 (D) ֒→ H −1 (D) and all the conditions for existence and uniqueness of solution in [16] are satisfied. Thus (7)  We now prove an estimate similar to (3) for the solutions of (7). We will do this by applying the Itô formula from Krylov [14]. To that end we need to consider the functions We now collect some key properties of these functions. We see that φ n are twice continuously differentiable and where N depends on p and n ∈ N only. Further, for any r ∈ R, as n → ∞ and where N depends on p only.
These inequalities along with Young's inequality imply, for any ǫ > 0, where the last inequality is obtained using Hölder's inequality and N depends only on d, p and ǫ.
Using Theorem 3.1 from [14], we get that almost surely for any t ∈ [0, T ] and n ∈ N. Thus using Assumptions A-1, A-2 and Young's inequality for any ǫ > 0, we obtain almost surely for any t ∈ [0, T ] and n ∈ N. Here the generic constant N depends only on d, K and ǫ and Further, using Burkholder-Davis-Gundy's inequality, Remark 3(c) and Hölder's inequality, we see that which, using the same steps as before, in particular Remark 3 points (ii) and (iv), gives The next lemma follows from Lemma 3.3 in [6], however we include the proof for convenience of the reader.
where N = N (d, K, κ, p) and C m := E T 0 Proof. From (10) and Remark 3(iv),(v) and Assumption A-3, we get where N = N (d, p, K, ǫ) and Applying Gronwall's lemma, we obtain for any t ∈ [0, T ] Further, taking the supremum over t ∈ [0, T ] in (10), using the same estimates as given above and then taking expectation, we get using (11) where N does not depend on n and m. Thus, we have where N = N (d, p, K, κ, T ). Now we let n → ∞ and apply Fatou's lemma to complete the proof.
We can now use Lemma 2 and the monotonicity of r → f m t (x, r, z) to obtain an estimate for u m t , where the right-hand-side no longer depends on m. Let Proof. From (10) and Remark 3(iv), we get where N = N (d, p, K, κ). Taking limit n → ∞ and using Lebesgue's dominated convergence theorem in view of (12), (8) and (9), we get Using the fact rf m t (r, 0) ≤ 0 for any r ∈ R, m ∈ N, t ∈ [0, T ], Young's inequality and Assumption A-3, we get Substituting this in (15) and then applying Gronwall's lemma, we obtain for any t ∈ [0, T ] Further, taking the supremum over t ∈ [0, T ] in (10), using the same estimates as given above and then taking expectation, we get using (11) where N does not depend on n and m. Taking limit n → ∞ using Lebesgue's dominated convergence theorem and using (13) along with the steps as above, we get and hence the lemma.
To complete the proof of Theorem 1 we need to take the limit, as m → ∞ in (14) and to show that (1) has a solution. To that end we obtain the following result.
Therefore we may use Lebesgue Dominated Convergence Theorem to obtain Proof of Theorem 1. In the case when f t (r, z) is bounded (i.e. the case of α = 1 in A-3) the existence of unique L 2 -solution follows immediately from Krylov and Rozovskii [16] and the required estimates from Lemma 2. So we need to consider the case α > 1. In order to show the weak limit u obtained in Lemma 4 is indeed the unique solution of SPDE (1), it remains to show that f ′ = f (u, ∇u) which can be shown using the monotonicity argument as below.
Define for each w ∈ L α (D) ∩ H 1 0 (D), s ∈ (0, T ) and k ∈ N, the operators A s w := L s w + f 0 s and B k s w := M k s w + g k s .
We now need to re-arrange the right-hand side of (21) so that we can use the monotonicity assumptions. We have Using (18) and (19), we have and hence using (17) in (22) together with (21), we obtain for all t ∈ [0, T ] Now, integrating over t from 0 to T , letting m → ∞ and using the weak lower semicontinuity of the norm, we obtain where we have used Remark 4 in last inequality. Again, integrating from 0 to T in (20) and combining this with (23), we get which on using (17) gives Let η ∈ L ∞ ((0, T ) × Ω; R), φ ∈ C ∞ 0 (D), ǫ ∈ (0, 1) and let ψ = u − ǫηφ. Then from (24) one obtains that Dividing by ǫ, letting ǫ → 0, using Lebesgue dominated convergence theorem and Assumption A-3 leads to Since this holds for any η ∈ L ∞ ((0, T ) × Ω, P; R) and φ ∈ C ∞ 0 (D), one gets that f (u, ∇u) = f ′ which concludes the proof.
Further, taking m → ∞ in (14) and using the weak lower semicontinuity of the norm, we obtain the following estimates for the solution of (1)

INTERIOR REGULARITY
In this section, we present the results on interior regularity of the solution to SPDE (1). The main result is stated in Theorem 2. The idea is to prove the result for the linear SPDE first and then use it along with the L p -estimates obtained in Section 2 to prove Theorem 2. We do not claim the result for the linear case to be new, however we could not find such result in literature in sufficient generality.
To raise the regularity of the solution one needs the given data to be sufficiently smooth. Thus, we assume the following condition on the coefficients before stating the main result of this section.
One can obtain regularity results up to the boundary in appropriate weighted Sobolev space using results from Krylov [15] along with the L p -estimates obtained in Theorem 1. However, obtaining the similar results for the linear equations using L p -theory is more useful . We will discuss this in Section 4.
As mentioned before, we will first get the results for linear equations. So, we consider the following linear stochastic evolution equation: where the operators L and M k are defined in (2). As can be seen in what follows, one can raise the regularity to any order for the linear equation by assuming the given data to be sufficiently smooth. Thus we make the following assumption on initial data and the free terms and then state the result in Theorem 3. Let n ≥ 0 be an integer.
Lemma 5. Assume that v ∈ C([0, T ]; L 2 (D)) a.s., v is adapted and satisfies (26) and moreover v ∈ L 2 (Ω × (0, T ), P; H 1 (D)). If Assumptions A-2, A-5 and A-6 hold with n = 1, then Proof. We consider a cut-off function η ∈ C ∞ 0 (D) which is 1 on D ′ . Define the l thdifference quotient, l ∈ {1, 2, . . . , d}, by where T h l u(x) = u(x + he l ) is the shift operator and the step-size h satisfies 2|h| < dist(supp η, ∂D). From (26), we get Applying Itô's formula for the square of L 2 -norm, we get Note that operators δ h l and ∂ j are linear and hence they commute. Thus, using integration by parts and the formula where, where, Now extending η, f, g and v to R d by setting them to 0 on R d \ D and using the fact that Substituting (31)-(32) in (29), we get Further, it can be seen that the process M h t defined in (28) is a local martingale where a localizing sequence of stopping times converging to T as n → ∞ is given by Thus, replacing t by t ∧ τ n in (33), then taking expectation and choosing ǫ > 0 small enough such that 2κ − ǫC K,η = C κ > 0 and finally using Fatou's lemma, we get Using the inequalities of Burkholder-Davis-Gundy, Hölder and Young together with the estimates above we get that Replacing t by t∧τ n in (33), taking the supremum over t ∈ [0, T ] and using (36) we obtain which, on applying Fatou's lemma, yields where N = N (K, d, η, ǫ). Now note that the right hand side of above equation and (35) are independent of h and are finite and hence using e.g. [8, Ch. 5, Sec. 8, Theorem 3]), we get (27).
Proof of Theorem 2. Let u be the solution to (1) given by Theorem 1. Then considering f t (u t , ∇u t )+ f 0 t as a new free term f t , we observe that u satisfies (26) with such free term. Now under the Assumptions A-3, A-4 and due to Theorem 1, applied with p ≥ 2α − 2, we get the estimate (3) and hence Hence we can apply Theorem 3 with n = 1 thus proving the first claim. Moreover if f is a function of t, ω, x and r only such that (25) holds, then taking f t (u t ) + f 0 t as a new free term f t , similarly as above, we get ). Thus all the conditions of Theorem 3 are satisfied for n = 2. This yields the second claim.

REGULARITY IN WEIGHTED SPACES USING L p -THEORY & TIME REGULARITY
In this section, we raise the regularity of the solution to the SPDE (1) using L p -theory from Kim [12]. The reason for using L p -theory is that one gets better estimates for the solution of the corresponding linear equation, see Theorem 4, given below, which follows immediately from Kim [12,Theorem 2.9].
We will use this together with the L p -estimates we proved in Theorem 1 to obtain regularity results (both space and time) for the solution of the semilinear equation (1), see Theorems 5 and 6 below. In particular we obtain Hölder continuity in time of order 1 2 − 2 q for the solution to (1) as a process in weighted L q -space, where q comes from the integrability assumptions imposed on the data.
First, we introduce some notations, concepts and assumptions from Kim [12]. For r 0 > 0 and x ∈ R d , let B r0 (x) := {y ∈ R d : |x − y| < r 0 }. Definition 2 (Domain of class C 1 u ). The domain D ⊂ R d is said to be of class C 1 u if for any x 0 ∈ ∂D, there exist r 0 , K 0 , L 0 > 0 and a one-one, onto continuously differentiable map Ψ : B r0 (x 0 ) → G, for a domain G ⊂ R d , satisfying the following: Let D be of class C 1 u and ρ(x) := dist(x, ∂D). Then, by [12, Lemma 2.5], there exists a bounded real valued function ψ defined onD satisfying for any i = 1, . . . , d and any multi-index γ. Further it follows from Remark 2.7 in [13] and from the boundedness of D that for some constant In other words, ψ and ρ are comparable in D, and in estimates they can be used interchangeably (up to multiplication by a constant). Moreover this implies ψ ≥ 0. For 1 ≤ q < ∞, θ ∈ R and a non-negative integer n, define the weighted Sobolev space H n,q θ (D) by H n,q θ (D) := {u : ρ |γ|+(θ−d)/q D γ u ∈ L q (D) for any |γ| ≤ n} where the norm for u ∈ H n,q θ (D) is given by For functions u : R d → R d ′ , we define the norm analogously and use the same notation.
The following result from Lototsky [17] plays an important role in proving our results.
for all t and x, (iv) and for almost every (t, ω), the coefficients a ij (t, x) and σ i (t, x) are uniformly continuous in x ∈ D.
Note that, the operator L given by (2) is in divergence form but the results from [12] are for operators in non-divergence form. One knows that (1) can be expressed in nondivergence form if the coefficients a ij are differentiable. Thus Assumption A-7 implies Assumptions 2.2 and 2.3 in [12]. Hence the following theorem follows from Theorem 2.9 of Kim [12].
Proof. First we note thatθ > d and D is bounded, therefore sup x∈D ρθ −d (x) < ∞. Using this along with Assumption A-3 implies which is finite in view of Theorem 1 and the fact f 0 ∈ H 0,q θ (D). Now note that ψ is bounded onD and hence Proof of Theorem 5. Let u be the solution to (1) given by Theorem 1. Then considering f t (u t , ∇u t ) + f 0 t as a new free term f t , the solution u satisfies (40). We wish to apply Theorem 4 with n = 0 and in order to do so we need to show that ψf ∈ H 0,q θ (D). Indeed this follows immediately by using Lemma 7 withθ = θ andq = q. Hence applying Theorem 4 with n = 0 we obtain ψ −1 u ∈ H 2,q θ (D). This completes the proof of the first statement of the theorem.
We now consider the case when Assumption A-7 holds with n = 1. Again we will apply Theorem 4 (but now with n = 1 and q 2 in place of q) and so we need to show that ψf ∈ H 1,q θ (D) withq := q 2 . Takingθ = θ andq =q in Lemma 7, we get ψf ∈ H 0,q θ (D). Thus we consider Clearly I 1 < ∞ using (38), the fact ρ is bounded on D and Lemma 7 (withθ = θ and q =q). Further observe that where ∇ z f t is the gradient with respect to z of f t = f t (x, r, z). Thus, we have where, Now, using the fact that ψ and ρ are bounded on D and the assumption on growth of derivatives of the semilinear term, see (41), we observe that This is finite in view of Theorem 1, see the estimate (42) for details. Further, using Young's inequality and the fact that ψ and ρ are bounded on D along with growth assumption (41), we get